One way to highlight the arbitrariness of the standard classification is on formal grounds. Note that the mental state of desire is part of objective reality, and whether it qualifies as 'fulfilled' is a matter of fact no less objective than the question whether some belief qualifies as 'knowledge'. But a theory according to which knowledge is intrinsically valuable is typically considered a form of objectivism. That is: there can be 'objectivist' theories of value according to which mental states X are intrinsically valuable. The theory might even be monistic, and claim that X is the only thing that's intrinsically valuable. (This wouldn't be very plausible in case of knowledge, but for all its implausibility, such a view would clearly be objectivist in nature.) It would seem inexplicably strange if objectivists were barred from substituting "fulfilled desires" for X -- a mere change in content. Presumably we want the distinction between 'objective' and 'subjective' theories to track some deep difference in form.
Note that someone might hold that fulfilled desires are objectively valuable in just the same way that other objectivists hold that knowledge (say) is objectively valuable. This seems especially clear in the case of preference utilitarians who claim that we have reason to satisfy other people's preferences even if we don't want to. (The fact that we don't want to is itself a reason that counts for something. But so is the other person's desire, and it might outweigh our own.) The relevant feature here seems to be that we are normatively bound by a higher authority than our contingent, immediate perspective. This may be true even on desire-based accounts, if they impel us to respect other desires besides our present ones -- i.e. other people's desires, or even just our own future desires, for that matter.
A sure sign of value objectivism, as I've defined it, is when a theory imposes on us alienating aspirations -- a condition certainly satisfied by any form of utilitarianism, including desire-based versions.
Value subjectivism, by contrast, insists that the values or normative reasons to which one is beholden must be firmly rooted in one's current sentiments or 'deliberative standpoint'. Subjectivism thus entails some kind of 'present aim theory' (in Parfit's terminology), allowing only what Bernard Williams calls 'internal reasons' -- reasons that can gain traction and motivate us given our actual psychologies.*
* (But what if our actual psychologies are irrational? Isn't some idealization needed here? But then, if we build enough into our understanding of 'rationality' -- cf. Kantians -- then the results may be just as alienating as before. I'll put aside such complications for now, but comments are welcome.)
I guess one could, as a matter of form, hold even the present aim theory in an 'objectivist' way. That is, one might just insist that objectively, "from the point of view of the universe", what's advisable for each person is that they fulfill their own present aims. So what's distinctive about subjectivism is not the content of our normative reasons, but the underlying explanation why we have the reasons that we do. Subjectivism appeals to our subjective perspectives as bedrock. Objectivists, if they appeal to our subjective perspectives at all, do so on the basis of some further, underlying consideration: e.g. that subjectively satisfied people constitute a better world.
So, even if satisfying desires is what finally matters (practically speaking), it is an open question why this is what matters, or what the source of this normativity is. Desire-based subjectivism will ground the putative normativity of desire in the agent's own (unalienated) perspective, whereas desire-based objectivism will ground it in something larger than the agent -- possibly, the universe itself. This, I propose, is the most theoretically interesting way to divide 'subjective' and 'objective' theories of value.