I think you're doing something interesting here, but you need to be more careful to distinguish between moral epistemology and moral metaphysics. And the word "objective" is not always your friend in making this distinction.
Let's go through it piece by piece:
Moral nihilists argue that moral standards cannot be objective to the extent that moral truths do not exist;
First off, you are right that, definitionally, moral nihilists do not believe there are moral truths. But it does not follow that on such a view "moral standards cannot be objective."
There's at least two reasons why. First, we need to consider the work done by the term "moral standard". If this merely means "moral truth," then we don't need it in the claim. If it means something more, then it would seem it means something like the standards that a society claims are or are built on moral truths. A moral nihilist needn't have a commitment as to what "moral standards" qua social entities are and how they work except that they clearly have no basis in truth.
Second, we need to think carefully about the use of the word "objective" to see what sort of work it's doing for us and whether it's the best word for that. Or maybe to put it another way, are there sense of objective that nihilist can accept vis-a-vis these standards? My answer is yes -- depending on what we mean by "objective." One meaning of "objective" stands in contrast to the idea that "subjective" means non-transferable from the subject. Or "objective" = out there in the shared world. But there's no reason a nihilist cannot accept that "moral standards" are out there in the shared world -- or even that they are held by everyone in the world. (There's also several other senses of "objective" that can be added). Consequently what I'd suggest is that you mean "universal" or "universally valid" giving something like this:
Moral nihilists insofar as they deny moral truths exist also deny that universal moral standards exist.
Moving to the second half:
the existence of moral truths would imply these are objective, and thus to deny the existence of objective moral standards a moral skeptic must deny the existence of moral truths.
I think here again we need to be careful about "objective." One meaning of objective is that everyone could know them or they would not vary by the person. As Dennis' answer mentions moral subjectivists believe moral truths exist but are subject dependent, so it seems false to suppose that the existence of moral truths implies that they are objective.
But perhaps you could argue that the moral nihilist and the moral realist are kin in sharing an assumption that moral truths would be universal and accessible if they existed the same way that libertarians (about free will) and hard determinists are kin in sharing an assumption about the nature of free will but taking the pro and con sides.
I still think this is not ideal, however, for a second reason. Here, I think we need to distinguish metaphysics and epistemology. Whether a moral truth exists is a metaphysical question; whether anyone knows it is an epistemological question. The word "objective" seems to relate more to the epistemological axis then the question of existence.
Most (all?) moral realists also believe these truths can be known. But the distinction between moral skeptics and moral nihilists is that the former attack our ability to know and the latter the existence of moral truths themselves.
To push further, the moral nihilist does not need to deny the moral standard is objective -- instead, the moral nihilist is denying that "the moral standard" is moral, because morality does not (he or she contends) exist. The nihilist can accept something people call a "moral standard" exists and that it is "objective" in at least some sense, but they would reject that it has any connection to moral bedrock -- because qua moral nihilist they don't think such bedrock exists.
In my view, you'd be better off avoiding the word "objective" and also focusing carefully on distinguishing the metaphysics and epistemology.