Easiest thing is just to ask someone. Customs obviously vary, so which type won’t tell you what you should do. Easiest thing there is also just to ask someone.
If you’re still not sure whether to wear it, wear it. You can’t offend anybody by wearing it, and it’s possible that you might offend by not doing so.
I’m not at all religious, but I do like tradition, and I appreciate the gesture. That said, it seems really draconian for a synagogue to require all men to wear one. If a guy really prefers not to wear one, he should still be welcomed. Respect goes in both directions.
My understanding re men covering their heads in the synagogue and uncovering them in church/indoors is that in the medieval west, uncovering your head was a sign of respect (removing the helmet of your suit of armor to show you were a friend), but in the ancient Middle East, the sign of respect was for men to cover their heads. Someone correct me if I’m off base (I know you will).
OK, that’s pretty much what I’ve been doing, as mentioned above. I was just wondering if there was any way of knowing otherwise. I personally don’t mind being required to wear a kippah when asked/required to. Other religions have other customs (like taking off shoes when walking in certain areas, and things of that nature) that I follow despite not being religious, but I’m not the sort of person to make waves in these kinds of cultural situations. I figure “their house, their rules.”