I agree with the PPs that "Fake Christmas" can be a hurtful term. But, I'd also encourage a conversation with MIL about how the Christmas traditions are evolving.
I would also encourage you to look at it from MIL's point of view. While you're celebrating Christmas at home with your parents and family, she is having an abbreviated Christmas day celebration, because her family celebration has been pushed back a few days. Sure, it's just a day, but if she's always had the big family celebration on Dec. 25, it's difficult to let that go. I would definitely encourage her to do the big family celebration on Christmas day with those who are in town to celebrate it, and have the abbreviated celebration when you arrive. Sure, it's a bit awkward to be the only family opening a pile of gifts, but you get through it. FWIW, we live that scenario each year, because my family is out-of-town and we, too, have opted not to leave our hometown on Christmas day.