It was the ambiguity of the word 'web' that tripped me up. You still need a network of computers for email to be useful.
`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.
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.