Turned down a yarmulke at a classmate’s Bar Mitzvah, was that rude of me?

In 6th grade my classmates were all invited to one kid’s bar mitzvah. For the service at the synagogue all the boys were offered a yarmulke at the entrance. I turned it down, figuring that as a non-Jew it would be improper of me to wear it. Given that most of us kids were not Jewish I should have realized that since all the boys were being offered them it was probably the polite thing to do to wear it but I was 12 lol 🤷‍♂️. At some point I realized I was the only male not wearing one and felt a little self-conscious.

So was that rude of me? This was ages ago so there’s not much point in tracking anyone down to apologize, but I’d like to know in case I’m even at a synagogue again.

Also: I always thought the word for the Jewish scull cap was “yarmulke,” but I keep seeing them called”kippa” now, is there a difference or is this a recent linguistic change?