YANBU. Firm, but you gave her the option.
Incidentally a few years ago I heard about a couple who always agreed that if either parent said 'Little Johnny, do that again and we will cancel, turn around, stop XY or Z' the other would absolutley back them up regardless.
So they were driving across the US to get to one of the Disneys.. a good few hours drive for them, everything booked, just waiting for them to turn up and have their weeks holiday.
The DC were playing up badly in the car. The mum said, 'Kids if you don't stop, we'll turn the car around and go right back home.' Last chance.
The kids carried on playing up, from what I remember it was pretty bad fighting, screaming, full on and unwarranted bad car behaviour.
So Mum said, OK Dad, turn the car around, the holiday is cancelled. and he did.
The kids sat in shocked silence the whole way home, and never again played up in the car, and never again ignored their parents 'last chance' requests.
They spent the week at their home, lost the money for the holiday, and the parents never once regretted it.
When we say something, we HAVE to mean it, especially at 8.. that's old enough to know how to behave, and far too old for tantrums in shopping centres.
If a few more parents drew the line as you have done, then we wouldn't have so many issues with kids behaving abysmally outdoors.
Actions and consequences.
An invaluable life lesson.