Moving from downtown SF to a suburb of SF does not count as "leaving San Francisco".

You're still using the same airport. You're still able to have in-person meetings with the same people. Travel time from SF downtown to your new place is significantly less than between two opposite ends of Beijing or Seoul.

Does anyone claim that?

Most people say Bay Area

Two examples that come to mind:

https://www.sfchronicle.com/business/article/2nd-most-valuable-U-S-startup-to-leave-SF-as-14558067.php

https://twitter.com/amasad/status/1785125298398446016

I read both of these as a vote against SF city governance and dumb policies (eg tax on GMV not revenue) and a vote for the hard to kill network effect of Silicon Valley.

You can argue that they do enough to clarify where they're moving to, but I think the overall tone in both cases is waaaaay more bombastic than relocating an office by 10-40 km justifies, and so to someone not intimately familiar local context/geography it sounds like they're actually moving cities.