It started off as people who are involuntarily celibate due to unfortunate circumstances. It ended up being co-opted by individuals incapable of self-reflection and self-improvement to understand that their celibacy is entirely based on their lack of hygiene and attitude towards women. They would rather blame the world for their problems instead of owning up to them and making positive changes in their lives.
Does identifying as an incel automatically come with those negative beliefs around gender and society, or should those two have separate terms?
Absolutely, yes. Incel culture is one of misogyny and sexism. People identifying as incels are typically the ones who blame women for not wanting to have sex with them while doing nothing to make women actually want to have sex with them.
My advice to you OP is stay far away from individuals and communities identifying as incel or pro-incel.