If so, which site that I use updates the "seen X mins ago"? All? Only the one I actually used to log in originally?
I was planning on testing this myself, but realized there is no way to check that value without updating it.
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Yes, it does, as per this answer on a question for the Android app.
We're now counting access to a site via the app for both "last seen" and the "daily site access" for the Enthusiast and Fanatic badges.
As is true with the sites, we don't actually count all access; only those actions we consider "real". For the mobile app, this basically means accessing the feed doesn't count; we'll probably tweak what we consider "real" over time.