For me, Europe is a geographical term more than anything else. The excerpt above that starts with "Geographically Europe is a part of..." best describes Europe as such.

This is true for me too - and when I read "center of Europe", I can only think of the geographical sense of "Europe". BTW, the Southeastern geographical border ambiguities shan't change the center much - the Caucasus Mountains - Kura river difference is only a 100 km wide strip, the Ural-Emba rivers idfference is roughly 500 km x 500 km, altogether a difference of 0.33 million km² - in a 10 million km² area.

You can visualise it on another relevant map from the Wiki article (here the Emba and Kura rivers are the borders; Georgia's and Azerbaijan's northern border is the Caucasus Mountains, the Ural river follows Russian-Kazakh border half-way East and then turns South):

II Europe, according to one commonly-reckoned definition
II  Extension over Asia of the continuous territory of a European state
II Geographically in Asia, considered European for cultural and historical reasons

_{*Lunatic*, n.
One whose delusions are out of fashion.}