Back in the old days, when computers were more expensive than a house, they were called mainframes. Computing was expensive, and still many people needed access. The solution? Terminals!
These were "dumb" computers, nothing more than a screen with a keyboard and a very basic computer. The terminal had a permanent connection to the mainframe. Commands were processed by the mainframe which returned results back to (the display of) the terminal. Most of what you did was text based, no graphics, no mouse, so all the terminal had to do was send and receive text, and display it on the screen.
If you take Chrome OS, or if you take a modern browser with a permanent internet connection and someone who does everything in the cloud, you get more or less the same idea. So your analogy of the browser is in fact very good.
Wikipedia says that a shell is a user interface to a computer, and it can be graphical as well. MS Windows is a shell. The UNIX or Linux shell is a CLI, commandline interface. Again, your browser analogy works here as well, with the webpage as the interface to the website and its database.