I don't see where the OP is embellishing anything.
I regularly see parents dragging small, unwilling children while I'm out and about.
Her youngest DD was screaming loudly enough for the OP to hear her and recognise her from the other side of the shop.
She saw a stranger dragging her DD along and although her older children have said they have met this woman once before, all three wanted to go home with the OP rather than stay with this woman or go back to Jane.
OP took them, which I think I'd have done myself in the same situation.
OP didn't go and find Jane, which with hindsight might have been the best idea, but in the midst of what she herself calls a bizarre situation, she perhaps wasn't thinking that clearly.
Also, if Jane was in the salon and all of a sudden her partner's ex turns up with their three kids and Jane's mate in tow, children all saying they want to go home with their mum, OP telling Jane she was taking them home, Jane's mate trying to give her side of things, that might actually have caused more trouble than just taking them and phoning her ex from the car.
My phone doesn't work inside big department stores, I can never get a decent signal. I wouldn't want to waste a low battery trying to get a signal from inside a shop, I think I might wait until we were outside or I could get to the charger in the car too.
I haven't seen one thing here that makes me think the OP has embellished anything. In fact I wonder if it's the other way, and Jane's mate has downplayed the tantrum to try to make herself look more reasonable than the OP.
If she's only met the children once before, and OP's DD does have "epic tantrums" which the OP says even her ex can't deal with very well, she was probably flustered, panicking, and likely to try and just drag her outside to get away from people staring. And then downplay how badly she handled it to Jane afterwards. I think that's much more likely than the OP embellishing anything here, because everything she has said sounds quite plausible really.