To start with, you're mixing up TCP and HTTP. Plain TCP sockets don't deal with HTTP URLs (or any other kind of URL really) – they're only a building block on top of which HTTP and similar protocols could be implemented – so if you're working with raw TCP, the http:// prefix has to go.
Second, bind() is an entirely local matter. It doesn't contact your DNS server, and it doesn't have any way to "pull" connections; it only starts accepting connections that would already arrive at your machine anyway.
This means that you need to manually update your domain to point to the right machine.
Third, most importantly, sockets are never actually bound to a domain, only to an IP address. Even though the TcpListener class might accept a domain name, what actually does is translate that domain to the associated IP address and bind to that address; e.g. when you specify "localhost:80" that just expands to "[::1]:80" plus "127.0.0.1:80".
So the first question is: What IP address does the domain name point to? And does that address belong to the machine you're running the code on?
If the IP address isn't assigned to the machine, then it doesn't make sense to bind a socket to it – the machine would never receive connections to someone else's IP address, so the OS won't even let you do that. In general, the OS won't accept any address that doesn't show up in ipconfig or ip addr, so e.g. if your machine is behind NAT with a private address, it'll only let you bind to its own private address (even though a domain name would point to the public address instead).
Because of this, it's generally rare and not very useful to specify a domain name to bind() even in programming languages that happen to accept it. Instead, specify the IP address manually – or use 0.0.0.0 as a shortcut to mean "all IP addresses". Then, as mentioned already, visit your domain's DNS control panel and make sure it points to your address.
(If it points at your web-host server and you're trying to run the Rust code at home, you can keep them separate by creating a subdomain.)