Although not as clear as I would like, this is a sort of working version to at least provide some customisation to error pages. It is a first approach but hopefully can help others.
The list of handled exceptions is not extensive, but mainly addressing 404 errors (NoHandlerFoundException) and other typical errors like InternalServerErrorException and NullPointerException, to try to catch them all in the end with a generic error for everything else that is an Exception).
Note that this is not covering other exceptions related to e.g. bad syntax in a JSTL template (org.apache.jasper.*; exceptions that cannot apparently be caught here).
These are the related changes and additions to the source base:
CustomSimpleMappingExceptionResolver.java (provide generic exceptions, yet logs details)
package ...;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Date;
import javax.ws.rs.InternalServerErrorException;
import javax.servlet.http.HttpServletRequest;
import javax.servlet.http.HttpServletResponse;
import org.springframework.http.HttpStatus;
import org.springframework.web.servlet.handler.SimpleMappingExceptionResolver;
import org.springframework.web.servlet.ModelAndView;
import org.springframework.web.servlet.NoHandlerFoundException;
public class CustomSimpleMappingExceptionResolver extends SimpleMappingExceptionResolver {
    public CustomSimpleMappingExceptionResolver() {
        // Turn logging on by default
        setWarnLogCategory(getClass().getName());
    }
    @Override
    public String buildLogMessage(Exception e, HttpServletRequest req) {
        return "MVC exception: " + e.getLocalizedMessage();
    }
    @Override
    protected ModelAndView doResolveException(HttpServletRequest request, HttpServletResponse response,
                                              Object handler, Exception ex) {
        // Log exception
        ex.printStackTrace();
        String exceptionCause = ex.toString();
        String exceptionType = ex.getClass().getCanonicalName();
        // Get the ModelAndView to use
        ModelAndView mav = super.doResolveException(request, response, handler, ex);
        // Make more information available to the view - note that SimpleMappingExceptionResolver adds the exception already
        mav.addObject("url", request.getRequestURL());
        mav.addObject("timestamp", new Date());
        ArrayList<String> exceptions404 = new ArrayList<String>(
                Arrays.asList(
                        NoHandlerFoundException.class.getName()
                        )
        );
        ArrayList<String> exceptions500 = new ArrayList<String>(
                Arrays.asList(
                        InternalServerErrorException.class.getName(),
                        NullPointerException.class.getName()
                        )
        );
        String userExceptionDetail = ex.toString();
        String errorHuman = "";
        String errorTech = "";
        if (exceptions404.contains(exceptionType)) {
            errorHuman = "We cannot find the page you are looking for";
            errorTech = "Page not found";
            userExceptionDetail = String.format("The page %s cannot be found", request.getRequestURL());
            mav.setViewName("/error/404");
            mav.addObject("status", HttpStatus.NOT_FOUND.value());
        } else if (exceptions500.contains(exceptionType)) {
            errorHuman = "We cannot currently serve the page you request";
            errorTech = "Internal error";
            userExceptionDetail = "The current page refuses to load due to an internal error";
            mav.setViewName("/error/500");
            mav.addObject("status", HttpStatus.INTERNAL_SERVER_ERROR.value());
        } else {
            errorHuman = "We cannot serve the current page";
            errorTech = "General error";
            userExceptionDetail = "A generic error prevents from serving the page";
            mav.setViewName("/error/generic");
            mav.addObject("status", response.getStatus());
        }
        Exception userException = new Exception(userExceptionDetail);
        mav.addObject("error_human", errorHuman);
        mav.addObject("error_tech", errorTech);
        mav.addObject("exception", userException);
        return mav;
    }
}
WebAppConfig.java (registers custom exception resolver as an exception handler)
package ...;
import org.springframework.context.annotation.Bean;
import org.springframework.context.annotation.ComponentScan;
import org.springframework.context.annotation.Configuration;
import org.springframework.context.annotation.PropertySource;
import org.springframework.context.annotation.Scope;
import org.springframework.context.annotation.ScopedProxyMode;
import org.springframework.core.env.Environment;
import org.springframework.core.Ordered;
import org.springframework.http.HttpStatus;
import org.springframework.transaction.annotation.EnableTransactionManagement;
import org.springframework.web.servlet.config.annotation.EnableWebMvc;
import org.springframework.web.servlet.config.annotation.ViewControllerRegistry;
import org.springframework.web.servlet.config.annotation.WebMvcConfigurerAdapter;
import org.springframework.web.servlet.HandlerExceptionResolver;
import org.springframework.web.servlet.NoHandlerFoundException;
import org.springframework.web.servlet.view.InternalResourceViewResolver;
import java.lang.ClassNotFoundException;
import java.lang.NullPointerException;
import javax.annotation.Resource;
import javax.ws.rs.InternalServerErrorException;
import java.util.Properties;
@Configuration
@ComponentScan("...")
@EnableWebMvc
@EnableTransactionManagement
@PropertySource("classpath:application.properties")
public class WebAppConfig extends WebMvcConfigurerAdapter {
    @Resource
    private Environment env;
    // ...
    @Bean
    HandlerExceptionResolver customExceptionResolver () {
        CustomSimpleMappingExceptionResolver resolver = new CustomSimpleMappingExceptionResolver();
        Properties mappings = new Properties();
        // Mapping Spring internal error NoHandlerFoundException to a view name
        mappings.setProperty(NoHandlerFoundException.class.getName(), "/error/404");
        mappings.setProperty(InternalServerErrorException.class.getName(), "/error/500");
        mappings.setProperty(NullPointerException.class.getName(), "/error/500");
        mappings.setProperty(ClassNotFoundException.class.getName(), "/error/500");
        mappings.setProperty(Exception.class.getName(), "/error/generic");
        resolver.setExceptionMappings(mappings);
        // Set specific HTTP codes
        resolver.addStatusCode("404", HttpStatus.NOT_FOUND.value());
        resolver.addStatusCode("500", HttpStatus.INTERNAL_SERVER_ERROR.value());
        resolver.setDefaultErrorView("/error/generic");
        resolver.setDefaultStatusCode(200);
        // This resolver will be processed before the default ones
        resolver.setOrder(Ordered.HIGHEST_PRECEDENCE);
        resolver.setExceptionAttribute("exception");
        return resolver;
    }
    // ...
    @Bean
    public InternalResourceViewResolver setupViewResolver() {
        InternalResourceViewResolver resolver = new InternalResourceViewResolver();
        resolver.setPrefix("/WEB-INF/views");
        resolver.setSuffix(".jsp");
        resolver.setExposeContextBeansAsAttributes(true);
        return resolver;
    }
    @Override
    public void addViewControllers(ViewControllerRegistry registry) {
        super.addViewControllers(registry);
    }
}
Initializer.java (adding dispatcherServlet.setThrowExceptionIfNoHandlerFound(true);; maybe not needed)
package ...;
import org.springframework.web.WebApplicationInitializer;
import org.springframework.web.context.ContextLoaderListener;
import org.springframework.web.context.support.AnnotationConfigWebApplicationContext;
import org.springframework.web.servlet.DispatcherServlet;
import javax.servlet.ServletContext;
import javax.servlet.ServletException;
import javax.servlet.ServletRegistration;
public class Initializer implements WebApplicationInitializer {
    public void onStartup(ServletContext servletContext) throws ServletException {
        AnnotationConfigWebApplicationContext ctx = new AnnotationConfigWebApplicationContext();
        ctx.register(WebAppConfig.class);
        servletContext.addListener(new ContextLoaderListener(ctx));
        ctx.setServletContext(servletContext);
        DispatcherServlet dispatcherServlet = new DispatcherServlet(ctx);
        dispatcherServlet.setThrowExceptionIfNoHandlerFound(true);
        // Add the dispatcher servlet mapping manually and make it initialize automatically
        ServletRegistration.Dynamic servlet = servletContext.addServlet("dispatcher", dispatcherServlet);
        servlet.addMapping("/");
        servlet.addMapping("*.png");
        servlet.addMapping("*.jpg");
        servlet.addMapping("*.css");
        servlet.addMapping("*.js");
        servlet.addMapping("*.txt");
        servlet.setLoadOnStartup(1);
        // ...
    }
}
Structure of views and tags related to error classes:
    src/main/webapp/WEB-INF/
    ├── tags
    │   └── error.tag
    └── views
        ├── error
        │   ├── 404.jsp
        │   ├── 500.jsp
        └────── generic.jsp
src/main/webapp/WEB-INF/tags/error.tag
<%@taglib uri="http://java.sun.com/jsp/jstl/core" prefix="c" %>
<!DOCTYPE html>
<head>
    <title>Error page</title>
</head>
<body>
<div class="container">
    <h3><c:out value="${error_human}" /></h3>
    <p><br/><br/></p>
    <div class="panel panel-primary">
        <div class="panel-heading">
            <c:out value="${error_tech}" />
        </div>
        <div class="panel-body">
            <p><c:out value="${exception_message}" /></p>
        </div>
    </div>
</div>
</body>
</html>
src/main/webapp/WEB-INF/views/error/404.jsp
<%@ page language="java" contentType="text/html; charset=utf-8"
         pageEncoding="utf-8" isErrorPage="true" %>
<%@ taglib uri="http://java.sun.com/jsp/jstl/core" prefix="c" %>
<%@ taglib tagdir="/WEB-INF/tags/" prefix="g" %>
<c:set var = "error_human" scope = "session" value = "We cannot find the page you are looking for"/>
<c:set var = "error_tech" scope = "session" value = "Page not found"/>
<c:set var = "exception_message" scope = "session" value = "The current page cannot be found"/>
<g:error />
src/main/webapp/WEB-INF/views/error/500.jsp
<%@ page language="java" contentType="text/html; charset=utf-8"
         pageEncoding="utf-8" isErrorPage="true" %>
<%@ taglib uri="http://java.sun.com/jsp/jstl/core" prefix="c" %>
<%@ taglib tagdir="/WEB-INF/tags/" prefix="g" %>
<c:set var = "error_human" scope = "session" value = "We cannot currently serve the page you request"/>
<c:set var = "error_tech" scope = "session" value = "Internal error"/>
<c:set var = "exception_message" scope = "session" value = "The current page refuses to load due to an internal error"/>
<g:error />
src/main/webapp/WEB-INF/views/error/generic.jsp
<%@ page language="java" contentType="text/html; charset=utf-8"
         pageEncoding="utf-8" isErrorPage="true" %>
<%@ taglib uri="http://java.sun.com/jsp/jstl/core" prefix="c" %>
<%@ taglib tagdir="/WEB-INF/tags/" prefix="g" %>
<c:set var = "error_human" scope = "session" value = "We cannot serve the current page"/>
<c:set var = "error_tech" scope = "session" value = "General error"/>
<c:set var = "exception_message" scope = "session" value = "A generic error prevents from serving the page"/>
<g:error />