One problem here is that there are many things that go by the name "nihilism." On the one hand, there are specific or thesis nihilisms (that is, nihilisms with respect to a particular concept or claim). One might be a nihilist about morality and claim that there are no moral truths. Alternatively, one might be a nihilist about truth and argue that there are no objective and/or absolute truths (or perhaps even no truths at all). There is also existential nihilism, which is the view that life has no intrinsic meaning. This list is not exhaustive.

Then there is what might be called movement or ideological nihilism, which is a broader worldview that embraces several (and, in more radical strains, all) of the specific nihilisms mentioned above and draws pessimistic conclusions about life, the universe, and everything from them. To hold one of the specific nihilisms, or even more than one, does not automatically make one an ideological nihilist. It all depends on how one reacts to the specific nihilisms one embraces. One need not be a pessimist just because one embraces a specific nihilism or two.

For example, a moral nihilist rejects the idea that there are moral truths but may yet think that we can nonetheless invent a replacement that functions as well or better than the traditional moral edifice. Similarly, one might accept the existential nihilism but reject the pessimistic conclusions that ideological nihilists typically draw from it (this being the characteristic feature of existentialism). Thus it would be a mistake to conflate the specific nihilisms with the broader ideological nihilism that some see as the logical consequence of those views.

The "destructive" part may derive from the political influence of nihilism, first in Russia in the 19th century and then elsewhere. Political nihilism seems to have involved the rejection of all authority. Bakunin is usually considered a nihilist from the political perspective. It seems to have become common to brand as nihilist--and so bomb-throwers and assassins--anybody radical or who violently objected to government or private property, whether they were or not, and especially if they were foreign (here in the U.S.). So, Sacco and Vanzetti were called nihilists, though it seems they were, if anything, men with anarchist leanings.