gavra_at_work,
So after all that misunderstanding that y’all agree that a skicap, football helmet, or baseball cap is sufficient
Not quite.
1. There is a concept in halacha of dressing nicely for davening.
2. In some communities, a hat and jacket are considered part of the requirement for dressing nicely for davening.
3. This does not mean that every hat and every jacket fulfill 1. As old man has repeatedly pointed out, a nice hat and jacket that are dirty do not fulfill 1. I would think that extremely casual or strange hats might also fail to fulfill 1.