Maybe we use language differently, but I would definitely describe your position as a belief system, and I wouldn't see that as any criticism. I wouldn't dream of attacking you for it, or looking for equivalent sneers on the lines of "god botherers" and "invisible friend".

Children are indeed taught that certain things are held by their parents and teachers are true. That applies to all kinds of things, as well as religion. As they grow older their ability to evaluate what they are taught develops, and good schools, including religious schools encourage this. Naturally enough, the physical environment in a school is likely to reflect its culture. A Catholic school will have Catholic symbols. In the same way, when my son was in the Woodcraft Folk, the song book contained The Red Flag.

People who are hostile to religion will also seek to communicate this to their children, I would suggest.

In both cases I doubt very much if adults feel compelled to follow their parents views.