It has to be that the this language was adopted as a compromise between being technically correct and non-tech-friendly. Most people aren't switching browsers on their same machine, so for non-tech-minded people, their "computer" is synonymous with their browser. Imagine what they'd have to say if they wanted to be technically correct? Something like "Can we store a cookie in the browser you're currently using so we can automatically sign you in the next time you visit this site?"?

Some sites like Peanutlabs uses Evercookie (in addition to a whole host of other incredibly abusive borderline malware techniques) to persist sessions, so.. it may apply sometimes.

Yeah, those solutions do work, but they are definitely borderline grey area and not always going to work on locked down machines (say in a corporate environment or public computer in a airport or library).