You're right that "log in" is more distinct from "sign up" (and works because "logging up" doesn't make much sense), and I definitely agree with you that 'login' is a noun, if anything!

Both "log in" and "sign in" are anachronisms though - they're justified when you're actually signing-in-and-out of a computer system or building to log your visit, but the metaphor doesn't technically make as much sense when you're talking about authenticating the user on a website.

The evolution of the terms does let you nicely abstract away the actual mechanism of authentication for a user, which I suppose is the point.