What? Scandinavia isn't part of Europe now? The "EU" is an organization, "Europe" is a continent. "European" includes everybody living in Europe (and yes, that even includes the Brits and most Russians).
I guess the confusion comes from thinking of "European" as some sort of cultural identity (which frankly doesn't exist, unless you only look at the last hundred years or so, and even that is generous), instead of the people living in the geographical region called "Europe".
Your 2nd point is spot on. This is where the cultural distinction matters.
By that definition India would be its own continent because the country largely sits on its own tectonic plate.
Also Europe and Asia are continents, they're not on the same continent (same landmass: sure).