As Nietzsche is an obvious example, I am focusing on him.
I think that there are no nihilist philosophers, because if someone is a nihilist, why would the nihilist even bother telling us? As a nihilist he must not care about anything.
Nihilism does not imply that "one does not care about anything". For that can fit into another definition : Apatheism (although apatheism means one does not care about God's existence or non-existence).
Anyway, Nihilists like any other philosophers, do actually care about purpose (or, in this case non-purpose).
It is clear that there is a metaphysical problem : one cannot prove / disprove the existence of purpose or meaning in the physical world.
To understand part of the problem, suppose that everything that exists has meaning, therefore it follows that everything X has a meaning and purpose on which it depends.
If x is the meaning/purpose of y, then y cannot contribute to the meaning of x, since this would be circular.
From this, one can infer that if everything has purpose / meaning, then everything is infinite, since every x needs an element y (as its source of meaning), so that x is not part of the set of all things that contribute to the meaning / purpose of y.
Which means that either beings are infinite, or the ultimate reality Y that contributes to the meaning/purpose of all existing and finite elements does not have itself meaning. So, everything has meaning/purpose, except the ultimate reality itself.
This is I think one of the best arguments for Nihilism, and I don't think Nihilism implies not caring about such and such.
Regarding my argument for Nihilism, that the ultimate reality does not have an intrinsic meaning or purpose to it. It is only a product of my analytical study of the subject, and I do not remember the source of this particular idea.
To read more about Nihilism and the meaning of life : https://plato.stanford.edu/entries/life-meaning
You are assuming that caring motivates action rather than constraint. People really aren't wired like that. It is generally harder to make a child not do something. Isn't it? (From the POV of Cognitive Dissonance Theory, we learn to be lazy to assert the value to our labor, which is caring about something.)
Our most famous nihilist argument is Montaigne, who in the end argues in favor of following tradition and letting everyone get used to it. If our emotions mean nothing, and they are all we really experience, we should just stop making ourselves miserable. Choosing some meaningless set of rules that keeps us from thinking about difficult things like morality is his personal prescription for not suffering. But he doesn't care whether or not you agree.
Classical Skepticism and Cynicism, on the contrary are not nihilisms. They have an agenda. (Or rather they have opposing agendas: the former of allowing for peace, the latter of disrupting undeserved peace.) They express a value. They are closer to Buddhism or the variants of Hinduism out of which it arose. There is a morality embedded in their disavowal of the meaningfulness of reality. And they wish to spread that message because they really think that morality is worthwhile. If we are deluded and suffering, we could stop. Letting us know that is a moral obligation.
From Montaigne's POV, then, why should one not meaninglessly give in to the frustration that no one ever sees this for what it is and they all sanctimoniously try to control everyone else? Then why not write a long annoying book about it? (Especially if it makes your mother feel better.) I see no contradiction there.
And Nietzsche does not qualify. He falls in the other category. He has a definite sense of value, even if it cannot be codified because it includes a devotion to originality and authenticity. It continually undercuts itself, but by his analysis so does every moral code. So why not cut out the middleman? He has an aesthetic ethics of opposing consistent moralities.
Nihilism is best understood contra the philosophy that it was/is trying to overthrow, this is the one predicated upon Christianity, and it's highest value, which is the Good. This is why of course Nietschze declared himself as the anti-Christ, though as someone who declares Christianity to be essentially a philosophy or religion of resentment, there is a great deal of resentiment in Nietzsche too.
It's worth noting that in a book of interviews with highly respected philosophers on the great philosophers, by Brian Magee, it was pointed that a good case can be made that intellectuals seduced by Nietzsche had a significant role to play in fascism - and this was by the way, by a philosopher sympathetic towards N's values.