I agree with others who have said it is time to politely tell MIL that it hurts your feelings when she refers to your celebration as "Fake Christmas." She probably thinks it's funny, and won't know you feel otherwise unless you tell her.
I do encourage you, though, to think about compromising just a little bit. While it's important to you that your child be home on Christmas Day, recognize that it's also important to your husband's family to see the three of you on the holiday itself. Perhaps you can go every third year, if you don't want to do every other year.
I have found through the years that my children are as excited to see what Santa brought to my parents' home or my in-laws' as they are when we're in our own home--the location doesn't matter to them at all. And while I agree with you that celebrating on the actual day shouldn't really matter, if my brother and his family refused to ever come on Christmas Day but insisted on only visiting at a time when I had to work and couldn't spend much time with them, I would be really hurt.