NFS: The Run also used it, Garden Warfare, Battlefield of course, Dragon Age: Inquisition and the next Mass Effect. I assume there are more that I haven't noticed yet.
I don't see why they'd change the name at all. Your suggestion does not make any sense to me.
Could it be because all the games you've named (even NFS The Run) include some form of fighting? Yes; I am specifically including NFS The Run's "vehicular combat modes", such as Battle Down The Mountain. Strangely, despite the engine's name, not all of the games have snow (or even cold weather), despite Garden Warfare's Yeti.