Email doesn't naturally involve the web? What?
Email predates the Web, and, imo, has been made much worse by all the Web-adjacent features shoved into it.
Email really is just a protocol for message sending, and it lives on it’s own port with its own server. If you have an email client and access to an email server (POP/SMTP/however), you can use email over the internet but without the “web”.
Basically, the web email client ought not be the only email client.
`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.
If the web didn't exist, which it didn't prior to 1991, email would still work fine. There just wouldn't be any web-based email clients.
The way we use email hasn't really changed that much since the 70s.