Are people from the U.K. considered Europeans? I know some Brits that get offended when they're called European, but that could be political.
Valid question that hasn't lost its relevance over the last couple of centuries. There's always been disagreement on both sides with support and opposition for both sides of the argument. During the period I study, quite a few German intellectuals explicitly argued, in reaction to the post-war wave of Anglo-American influence, that the Brits were not, and, instead, Germany should stick to it's continental roots. The same goes on the British side during the period, in which it wasn't uncommon for it to be argued that Britain should hesitate before meddling into European affairs. This is pretty clear at the outset of WWI. It was also a topic while good old Napoleon was running around the continent. Brexit and the reactions to it have made clear that discrepancies in opinion on this matter haven't lost their political significance.
"European" is a messy designation in general and the Cold War did an awful lot in shifting perceptions on who a European is and what type of European specifically. Germany was only considered Western after WWII, communist east Germany excluded, of course. Prior, they, along with Poland and Czech, were Central Europeans, the latter two of which are typically called Eastern European now (much to the chagrin of some Poles and Czechs). The EU, of course, throws another wrench into this, and the once seemingly likely prospect of Turkey joining doesn't complicate it any less.
In any case, I know some Scots and northern Irish who get offended by being called British while also identifying as European and wishing to stay within the EU.
tl;dr designations are never quite accurate and nobody will ever agree on anything.