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.
You're off base with your talk of terminals being "OS constructs" or whatever, I think it's because you're not being precise with what exactly you're talking about. "The terminal" is a combination of a terminal emulator, which is a userspace program which interprets control sequences that tell it which characters to render where. The terminal emulator is connected to a psuedoterminal, which is a pair of files for IPC with some special semantics, notably canonical mode. Importantly, in none of this is anything for controlling the system.
For controlling the system, you need some other program besides a terminal. A shell. A shell isn't part of the terminal, nor part of the operating system except in a broader sense of the term. And it doesn't have special abilities to control the system, anything the shell can do, a TUI or web interface, or any other kind of interface can be made to do.
Anyway, you ssh into a computer and start a TUI. The TUI doesn't render anything, it just sends you text and escape codes. The terminal emulator on your computer interprets this stream of escape codes and text, as a web browser interprets html, and renders it locally.