`Web`[0] is shorthand for `World Wide Web` which is specifically about HTTP/HTTPS and/or the applications built on that protocol. It is an entirely unambiguous word in this context.
`Internet`[1] is distinct, and that's the general purpose network of networks that you refer to which the Web is built on top of.
Totally fair! Frankly, I only know the distinction from a high school computers teacher who was adamant about the distinction.
I guess the easiest way to get the name is to see the “Web” as a “web” of hyper text documents, where hyperlinks act as the strands in the web (graph edges, if you will).
Honestly, like you say, it’s all built on top of a computer network (yet another web/graph). As a consequence, the distinction never really made a ton of sense to me, either.
Alas, this is the common parlance, so it is what it is.
