That's exactly what I have said the whole time looking
And in that sense, I agree with you. But the part of your posts that I was taking issue with, is that you have also been saying that Christmas isn't Christian. It is Christian.
I also think that the OP handled it badly, primarily in getting hostile straight away before discussing rationally and trying to find a compromise with the school, and in the double standard that actually some forms of Christian celebration are actually OK.
When push comes to shove, if you want to suggest who was most wrong, then the School were following the OPs written instructions. They legally had no choice but to do so. Yes, perhaps they could have found some middle ground, but at the end of the day, they had to follow the instructions that the OP gave them, which they did.
If, for example, I'd said to my school 'the children mustn't be allowed to buy sandwiches in the lunch hall - they need to choose a hot lunch', I would expect the school to to follow this instruction. If they'd come home and said something like 'I'm not allowed sandwiches at the picnic...' I'd think that the school were being a bit odd, but I would see how it's perfectly possible to interpret my instruction to follow this. Then I'd go the school and politely and rationally correct my error. I wouldn't blame them for following my instructions.