To play devil's advocate: to me, "logging out" and "exiting" are not the same thing. Particularly, the first is just a change of state--I'm still in your app, but now I'm no longer signed in. On the other hand, "exiting" is more like a motion; I expect to actually leave your app and not just be logged out. In UI terms, I see "exit" as a synonym of "quit".
Now, perhaps this makes sense for your app. However, if it makes sense to interact with it without being logged in, I think that is the wrong terminology. So I would definitely not use it for something like HN because browsing stories works regardless of whether I'm logged in or not.
I'm sure you thought about this far more when you were making your decision, so this is just my impression. You can always try to test the alternative empirically once you have a bunch of active users :).
We are looking into a few other alternatives including the most obvious where we convey that they are indeed leaving the app but in the next screen let the user know that their google account is still signed in and provide an option if they still want to sign out. Thankfully we have quite a while to launch, so I have time to run a couple other iterations and pick the one which seems most clear.
From what I understand it entered into English as a stage direction, but by the 16th century it had become a noun referring to a doorway for leaving.