Yes and no. The term 'Europe' is a lot of fun to discuss for this reason.
I reject the geographical definition, because it is self-evidently attempting to legitimize a cultural conception. Thus, I believe the borders of Europe are "fluid", and not something that can be defined in terms of mountains or seas - though there is a consensus that European colonies are not Europe. I would say that the geographic term of Europe is a limiting constraint - not all entities within these geographic borders are necessarily European, but all beyond it certainly are not.
Typically when somebody talks about European interests, there is an implicit [Western] involved, at least in my experience. I generally define "Europe", in the cultural sense, as roughly the territory in which Roman Catholicism, and later Protestantism, thrived, versus Eastern Europe, which would include the Byzantine regions and Russia. This is consistent with how the French and others used the term in the Middle Ages, to my knowledge, and how the Byzantines themselves used it (as they defined Europe in the same way the Greeks did, where Europe referred to northern Greece and lands beyond, but not Greece itself).
Here we can declare, at least, the unambiguous "core" of "Europe" - Poland & Scandinavia, being the latest additions, form the border of the "core", though this would expand along the Baltic northwards into Lithuania during the Teutonic crusades. Spain, or at least the areas under Christian control, could be called Europe - there was a French expression that said "Africa began at the Pyrenees" that support this cultural definition. Hungary would be the final "core" territory, bordering the Orthodox regions, but I would suggest this is open to debate.
The difficulty, as always, comes in defining the periphery states of Europe. Lithuania and the Baltics, up to Finland could be called periphery European states, as could Albania and Bosnia, for the same reasons. These areas were, at least initially, a mix of influences. The two most interesting cases to me are Hungary and Finland - they fit the religious definition, but there is an ambigious cultural overlap. I'd call them more "core" than "periphery" but less "core" than say, France or Italy.
The term European would then expand when the Turks and Mongols arrived. The Byzantines had a vested interest in claiming cultural ties with the "core" European states. The Russians did as well, but unlike the Byzantines, had less success in being viewed as European by "core" regions, at least initially. This is an expansion of the term European, but it still relies on religious identity - this time following Christ, though not necessarily the Roman Catholic Church. During this era, I think Albania and Bosnia establish themselves as unquestionably European.
The difficulty comes from Ottoman occupation, and conversion to Islam. Since the earlier definition of "periphery" was established to indicate a religiously mixed population, I think it can be re appropriated here. If we accept Europe as roughly synonymous with the term "Christendom", then we have a new periphery of states - Albania, Bosnia, and Russia [to a lesser degree today], due to their religious ambiguity, while formerly non-"European" states like Greece or Ukraine, which are unquestionably viewed as European today, maintain their status (though with an asterix of "Balkan" or "Eastern European" which pays homage to the initial definition based around the schism). There is also the issue of Georgia, and Cyprus. I would assign them qualities similar to Hungary or Finland above - more European than peripheral, but closer to the periphery than, say, Austria.
Armenia is a special case. I might call it peripheral, but this is due only to its geographic proximity to Georgia, as the new Christian definition certainly does not include the Oriental Orthodox communities, as evidenced by the unanimous agreement that Coptic Egyptians or Ethiopians are not Europeans. I think the case could be made, rather easily, that Armenia is not even a peripheral European state, for that reason, and simply a Christian Middle Eastern state.
Returning to Albania and Bosnia - I would call them peripheral European states, and consider their populations to be of peripheral European descent - those ambiguous regions where civilizations meet. But European itself is such a vague and changing term depending on where you are that I would hardly call my definition comprehensive or conclusive. I welcome debate.
What I do reject, however, is the idea that ambiguity surrounding a few cases nullifies the concept. Identity is not a concrete concept, certainly, but that does not mean it cannot be identified at all. In addition, while I do acknowledge the religious origins of the concept, I do not think it needs to maintain a religious aspect. European identity was formed at a time when religion was far more important that it was today. While I think Europeans can be secular, however, I do not think this can be inclusive of the faiths of other civilizations. Islam is not European, to give an example, though it has occupied "European territory". A secular Turkish society is still a society descended from a non-European identity, just as a secular French-society is still a society descended from a European one. The term "European" implies certain historical and cultural characteristics, and not just contemporary geographic or religious ones.
I also wouldn't claim European civilization is superior to others - it is merely different, and fear of racist intonations cannot leave us unable to define a concrete and observable historical identity.
It is a bit long, but what do you think?