> Modern art, as everyone should know, started in 1860s in Europe
As someone who is active in the German art scene, the understanding is not that clear-cut. Modern art, as I learned it, is "art in the modern period". So everything after the middle ages. Basically when people stopped only drawing religious art and portraits of nobles. What started in the 1860s is "modernism".
The term used more often (at least in Germany) is "contemporary art". Which translates to "everything after WW 2". I've seen a lot of contemporary art museums, but few modern art museums.