Not really, unless one accepts nihilism.
Not necessarily nihilism, unless one includes "eye for an eye" as a nihilistic belief.
I don't think so. I read a bit on WW 1 last year and I never got the sense that either was particularly motivated by Christian beliefs, although far as I understand both were Christians.
Their motivations could much more properly be attributed to nationalism, monarchy and/or ethnic separatism.
So, then, by this standard, motivation would be the primary determining factor on how they're labeled. If that's the case, then there can't be any such thing as "atheist terrorism," since atheism was not the primary motive of the regimes in question (as mentioned by the OP).
I would agree that nationalism was probably the primary motive for WW1, started with a bond between Serbian and Russian nationalism (linked by common religion and shared ethnic heritage) when it clashed with German nationalism (which was more economically motivated than religious).
Also, terrible as WW 1 was, I don't think either the Tzar or the Kaiser would qualify as terrorists of any kind to begin with. Their war declarations were very open and went through a lot of diplomatic effort.
War is not that much better than terrorism, but the distinction is still significant and worth of making.
The definition for "terrorism" has been manipulated and played around with so much, it's not really that useful in defining much of anything these days (i.e. "one man's terrorist is another man's freedom fighter"). In any case, I don't think open war declarations or diplomatic effort would exempt them from being considered terrorists, not by any standard I'm aware of. By that standard, even those on the OP's list of so-called "atheist terrorists" would not qualify as terrorists, since they also made open declarations of war and through a lot of diplomatic effort.
Even groups like the IRA and PLO tried to go the diplomatic route and were rather open in their declarations. So, if open war declarations and diplomatic effort are enough to define a group as "non-terrorist," then that would cover just about everybody.
I'm not even sure that I would agree that war is better than terrorism. War can be far more widespread, wipe out entire cities, destroy infrastructure and target entire populations. I've never lived under the occupation of a hostile foreign army before, but that sounds a heck of a lot worse than the possibility of a few ragtag terrorists lurking in the shadows.
It would not be atheist anymore, indeed. A "religion" is not a simple absence of atheism, however. There is considerable confusion running around about what atheism is and what it is not. Atheism may be a component of ideologies both good and bad (although it is far less dangerous in that regard than its opposite, theism), but it is not an ideology in and of itself.
I agree. I don't think beliefs are dangerous, in and of themselves. But there are true believers who think that theirs is the "one true way" and fanatically oppose anyone who doesn't believe. Then there are those who may or may not believe, but use the belief as a vehicle for some other agenda - such as any monarch or Pope who claimed to be Christian, but didn't act like one.
Indeed, some might argue that even Hitler or Stalin didn't actually believe in the ideologies they espoused, but just used the public's beliefs as a way of gaining political power. They would just as easily do away with a "fellow believer" if it meant more power, so it doesn't really matter what they actually "believe" in.
What makes them terrorists as opposed to, say, warmongers?
Do you feel that Christianity was a significant component of their motivation for military violence?
Well, again, it would depend on how one defines "terrorist."
I've noticed a lot of wordsmithing goes on regarding these terms, particularly if they end in the suffix "-ist." I'm not referring to you, but I've noticed those who seem to go out of their way and make great effort to want to make their labels stick. The label of "terrorist" and other such terms are designed to make others believe that "this is an evil person," and it's reflective of a political culture where soundbites and rhetoric prevail over reasoned discussion. Some people get rather upset and defensive if they're questioned or challenged on the terminology they use, and this could be indicative of a disingenuous motive behind the practice of labeling people or organizations without any coherent or consistent definition.