I have news for you -- the public barely understands webbrowsers -- mostly not.
Yes, we could take the time to explain that "the Internet" isn't just the web -- but you'll just lose your original point.
I'd rather accept "Mastodon" as a (maybe temporary) moniker while we bootstrap familiarity.
The worst that happens is people ask if something else is "compatible with Mastodon" which may end being a good baseline anyway if Twitter or another company try an Embrace-and-extend move