"Check in" is a triggered change in the Membership Command Center. When the inactivity timer triggers, you'll get a very eye-catching blue button that will be labelled "Check in". When you click it, you'll notice your "last activity" date stamp will update to the date you clicked the "Check in" button. So no need to worry, the timers run behind the scenes and trust me you'll notice when it happens.
Don't request a Re-invite unless you've been removed from either the main room (Gaiscioch or Gasiscioch na Murias) or your core room (Findias, Falias, Gorias) - you can check that by logging in to the game. If you're still guilded, you don't need a re-invite. That re-invite button is always visible, even when you're still guilded, so don't panic if you keep seeing it.
There are 2 timers in play, one for the main room (Gaiscioch or Gaiscioch na Murias) and one for your core room (Falias, Findias, Gorias). The main room has a lower inactivity trigger: 7 days website inactivity or 14 days game activity. The core room has a 45 day timer. If you time out of the main room, you can still chat in the core room while waiting for a re-invite.
Fog has put up a very nice post about it here; http://eso.gaiscioch.com/tavern/eso_discussion/post_67557.html where he describes the timers in a less confusing way than I do. xD