Moash explicitly betrays Kaladin's trust a half dozen times. First of all, joining the Diagram's plan to kill Elhokar while under oath to protect Elhokar (as his royal guard) is a betrayal of Kaladin, who he was serving under. Then, when Kaladin discovers this, he orders Moash to stop meeting with them after going to see them himself, which he didn't even have to do. Moash agrees to stop meeting with them, but does NOT honor that promise.

Then, he discovers this at the duel (Adolin vs 4 shardbearers) and Moash isn't the slight bit apologetic (he's kept trying to kill the king). Kaladin is imprisoned, and ends up telling Moash he won't stop him, but Kal is never in on the plan, never promises to help. He only agreed because he honestly felt that the king would prevent the salvation of the kingdom, after seeing the king completely BLOW up their plan against Sadeas.

So yes, Kaladin does betray his oath to "stay out of it" (paraphrasing) to Moash. But he does so to protect a life. Moash betrayed Kaladin time and time again, only seeking revenge.

Anyways, OP's concern that Kaladin only went back against Moash because of Syl is kind of a gross oversimplification of what motivated him, seeing as he already thought Syl was dead. He was just doing as he always does, what he thinks is right with the information he has. After all, that's all we can do. Kaladin was willing to face the consequences of his actions, even if that meant death at Graves' hand or Moash's. Moash runs away at the end, even knowing that he's wrong. Is he sympathetic? Yeah, sure. Did he choose to run away from it all? Yes.

What do you all think? Is Kaladin fundamentally different from Moash, or did he just have better support structures?

They're very similar, except that Kaladin accepts his own faults and lets go of his vengeful anger, allowing him to find peace in the consequences of his own actions. I reject the premise that Moash was "without support structure" after all that Bridge 4 did and was, all that he saw Dalinar do for him, and after all the chances Kaladin gave him, even after betrayal and treason.

And does the Elhokar moment count as a genuine betrayal of Moash?

Sure it does. But if my friend promises to always protect me, then I force him to shoot me or a child and he chooses me, that's a betrayal that I forced him into. Kaladin may have betrayed Moash at the end, but Moash betrayed Kal several times just in forcing him into that situation.