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?"?