What would a non-personal belief be?

There are beliefs that enough people hold that could be called a societal belief. For example: the US and freedom of religion, speech, etc, or the widely held beliefs by scientists on vaccinations. The right to hold and express those personal beliefs of religion, views on vaccination, etc, but not the right to affect others' lives with our own religion, our own non-vaccination.

Also you also can't even play that card, because LGBT beliefs for example are very personal.

Yes. And when an action or speech doesn't affect anyone else's personal life, it's fine. In the fight for LGBT equality, the personal beliefs of some (religion) are affecting the personal lives of others (marriage). When it becomes commonplace, LGBT marriages will never be mandated, will not be forced upon individual religions, etc.

these days people seem to think that sexuality/gender is determined by how you feel

Which people?

And what if a society happens to be mostly religious?

Is it all the same "flavor" of the same religion? If you want freedom of religion, then society has to agree to not impose a religion. Thus the societal belief of the US would be freedom of religion, and on the individual belief would be largely different flavors of Christian. Doesn't make it a Christian nation, makes it a nation with freedom of religion made up mostly of Christians.

Again, if religion isn't really defined,

As far as US federal law goes, it has been defined. As far as an easy layperson definition, I consider it anything in which literal worship is involved, with a set of ritualistic/dogmatic beliefs (usually written in a book of some sort) about or involving the supernatural.

Question: what definition would you use for religion to apply it to atheism?