Yes, it is absolutely the children's decision - not yours.
I'm with KoalaDownUnder on the 'proper' definition - replacement Mum. At very least, someone playing a significant maternal / caring role. I also recognise that the term is used far more widely and casually than that these days and that many people would mean nothing by it at all.
In my case, my Dad's Wife was the OW and my DM gained full custody. So, using a 'replacement Mum' label or anything that even hinted at that, would have been very hurtful. I was very protective towards DM so would have been upset by it too.
I was probably in my late 30s before I started to tolerate 'step mum' as a casual shorthand. We've always had a good relationship, even more so as adults, even after DF and 'wife' separated but on a 'name' basis and that's just fine.
Obviously nothing like your situation but, I'm making the point that children can have very real sensitivities about this, that words matter and, that children won't always speak up.
Therefore, it's up to them - and if other people start to use the term around them, they need to be asked what they'd prefer. They aren't necessarily going to speak up and they're socialised to acquiesce with adults, so need to be offered options, not presented with a loaded 'does this bother you?' sort of question.