There's also the issue that no one died to bring Beric back; he mentions seeing darkness, so I imagine the soul was just anchored and the body healed.
Cat's rez is closer to Drogo's with Mirri; Dany's son died to save her husband, and while he was alive, his mind was gone. Beric died to bring Cat back, but her mind remained instead of the body being healed.
I agree that LSH's craziness is partially attributed to her mental snap before her death, but I think it was also influenced by the idea that she literally has nothing to lose.
For me, undead vs living is an important distinction for where we go from here. I think Jon is gonna be a total badass, but if he's alive, there's probably less of a chance that he becomes the embodiment of full vengeance like LSH. I see him being more reckless though, or maybe giving less fucks, similar to how Beric took more chances after every resurrection.
Admittedly, Jon as a Coldhands type of monster to go against the White Walker King would be farkin amazing. I doubt that will happen though; even if he's not Azor Ahai, I think he needs to be the antithesis of death and darkness. A living, breathing Jon, resurrected by a priest of R'hllor accomplishes that nicely.