From what I’ve seen so far, my understanding is that when
Italy closes one eye, he’s usually telling both a truth and a lie; a half truth
if you will. It’s completely different from when he winks to tease or joke,
like you said.
Here’s an example:
When discussing the amount of work he does to his boss, Italy says that he doesn’t have a lot of spare time and that he’s proud of how quickly he completes work assigned to him. In a smaller blurb, he then sheepishly admits that it takes him a while before he gets around to it.

In the next panel, both eyes are closed as he blatantly lies
about why he lazes around during his afternoon breaks.
