Is vulnerable about letting people know how you feel or your weaknesses?

What about letting people know how you feel and your weaknesses while not caring if someone judges you for it? Is that being vulnerable or not?

I would say yes. Your weaknesses, if truly shared are weaknesses which can be used against you to hurt you and thereby you are vulnerable to them. Further, even if you don't care about the judgment of others then you can still be harmed by decisions of and social coordination between people who judge you.

We agree, assuming self knowledge, that the judgments of others tell you about them rather than about you.