I asked to learn using the other way, because I felt I had understood it better. They refused to show me the other way. I'm vaguely guessing that maybe they didn't know it and instead of explaining that insisted the other way was better.

Possible. I only learned one way and thought that was the one and only way.

I remember visiting friends of my father's and on the way out, three people (two from that family and I) were tying our shoes together, and I saw that each person was using a different method, which seemed really weird to me since I thought there was only one way.

So I can't tell you which way I do it since the method doesn't have a name for me beyond "tying my shoelaces" - the only variation is whether I do it once or twice. (I do it twice with my current shoelaces because they tend to come untied otherwise. Even so they sometimes do.) It might be what you call "bunny ears", though.