They are 2 different services, run by different people, one goes out of their way to bypass paywalls while the other doesn't, one is banned by Wikipedia and the other isn't, etc.

I think it's a distinction worth making.

Not to mention that the Wayback Machine itself isn't exactly a good tool to bypass paywalls as most paid sites don't let them archive paywalled content anyway.

Given that the root of the discussion is about Internet Archive being hit with huge traffic and not the functionalities provided by Wayback Machine, it very much is a distinction with a difference.

Yeah, you're mistaken. One archives web pages, the other maintains a list of paid-access accounts and fetches information from behind paywalls as a service.