The important detail isn't which exact term for Jewish scripture will most closely match the old testament in Christianity. The important detail is that "old testament" as a name is meaningless in reference to Jewish scripture, because the term only has meaning if you consider the new testament as equally valid scripture.
So they're arguing that referring to Moses in a Christian context is "appropriating" Jewish culture, while doing the exact same thing themselves in the exact same comment. If they actually cared at all they'd have known that using any Jewish name for the scripture would've served their point better than "old testament".
As for the Christmas thing, it doesn't make sense to call a Christmas movie Jewish because if you actually follow Judaism then the birth of Christ isn't something worth celebrating to you. Any Christ as the savior narrative goes directly against what Judaism believes about Christ. And any Christmas movie without Christ as a savior narrative, might as well be considered non-religious.
I didn't make any appeals to emotion, I just pointed out their own hypocrisy. If you want people to believe that you actually think Jewish culture is being appropriated by Christianity, the very least you need to do is not use Christian terminology when there exists widely known Jewish terminology for the same thing. If you don't know enough about Judaism to know the name Torah, then you have no right to complain about the interaction between Christianity and Judaism.
And of course Christianity believes that Judaism is unnecessary now. Just like Mormons believe the Christianity is unnecessary because they have the v3 update. It doesn't erase Judaism, heck the thing that started this whole thread was the fact that Jewish scripture is included directly in the Bible. The old testament stories are the same either way. The only difference is whether you believe that Jesus was the savior who fulfilled the law and brought the new law, or if you believe that the messiah hasn't come yet. But those stories still point to a future savior.