Plus if you're invisible no one can see when you've last been online. If someone saw you weren't online at all during the Night phase, s/he could conclude you aren't probably a gifted or a wolf, because they need to do things in the night-time, unlike ordos.
Oh, and never mind what I said about being inactive in Day1 before. I managed to mix up the days and thought Day1 would have been yesterday in RL...
