Yes, taking offence at the way other people describe themselves is down to their insecurity. Ridiculous!
The term stay at home mum exists only as a counterpart to the phrase work out of the home mum, both are such weird phrases. If you work, you can just say you work? Having to shoehorn 'mum' in there somewhere seems to smack more of insecurity than the reverse.
My DH is a not a work out of the home dad, nor is is he a 'full-time dad' and I doubt he could give a monkeys whether other dads describe themselves as such. Why is it only women that tie themselves in knots over these things? I think it really sets us back, all this weird infighting over terminology.
I am at home looking after children the hours of 9 and 5. I am not performing childcare duties between those hours, I am their mother. If you work outside the home you are not performing mothering duties between those hours, so why does it bother you that someone who is describes themselves as doing it full-time?
I have no vested interest in this btw. In my personal life if someone asks what I do, I say I'm at home with the children at the moment, and then get on with talking about something more interesting.
However, I have been in countless official situations in rl where someone has asked me what I do and whenever I've come up with some waffly response to attempt to describe it, they step in and go "full time mum". Because that is the reason I'm at home. To look after children full-time in my capacity as their mother. If you work you don't love your children any less than me, you are not any less their mother, but you're not wiping their arses while you're at work. It's short hand that everyone understands.
I can see if it's in your face on facebook with people saying "just love being a full time mummy to my wonderful kids" yes that's irritating. But that is really those people, and facebook that you loathe. It's not someone who just wants to make form filling an easier experience for themselves.