Within this thread, the OP and a few others make a good case, on paper, for the use of 'vaginal birth'. It makes sense. However, I don't think it matters. It's an emotive subject and many mothers care about not being made to feel 'less than', while others possibly enjoy feeling 'more than'.
I like to think I know what a mother means whichever and whatever terms they use, and if I don't, I'll ask a question (if appropriate at the time). Yes, there are many 'more correct' words to use and arguably words matter, but I would feel a total grunt bringing up this topic during a conversation with another mother or in a mums group. 'Don't you mean vaginally, because...?'
I don't think we are there yet societally. However, the PP who was told she was the only mother in the antenatal (?) class who didn't have a normal birth did not need to hear that! The lady clearly needed training.
Many women are already very sensitive about this topic much like they are when discussing breastfeeding v bottle feeding, or having one child over multiple, or ... etc.
I wouldn't feel comfortable discussing how I became a mother to someone I've just met (not that OP's advocating this). I do recall it wasn't straightforward! But if that person (usually a woman, right?) simply didn't ask other questions and seemingly wanted to just know whether it was vaginal, natural, normal or conventional over having an abnormal, unconventional, unnatural C-section, just so they can file it in some binary mental folder system of theirs, then I'd know enough about them to similarly file them in a 'less so' category.
Of course words matter, but in this context of women discussing the topic amongst each other, intention is important as well.