He's functionally right though, a remotely served interface can be implemented as a TUI behind ssh, or it could be implemented as a website. Which is the better option depends on the context, but they're both serving the same function, and the terminal emulator you're using to view the TUI is most certainly rendering it on your computer.

This is true only up to "you can send and receive data".

A terminal is an OS construct mostly. It will, with permissions, allow you to do anything the OS can do, including shutdown, or serve more data. It's the default mode of operating the system.

The opposite is not true. The default mode of operating a web browser will not let you create a terminal access or do anything new with the machine.

Insofar as the previous is false - because it website has flaws that allow code execution, shell escapes, or remote package installations, it's worth pointing out that these are usually possible because your web server is running as a user on a system, effectively with system access via similar interfaces as the shell.

This is like saying you can operate a car from the truck - yeah maybe through some weird gymnastics, but the trunk is for transfer, and the drivers seat is for operating.