How does one determine which are HP and which are morale? Can cure wounds both heal and inspire? If so, isn't that odd? I don't see anything about inspiration in the spell description. Am I misreading it?
While this thread is focused on 4E, IMO it really doesn't matter which edition you're talking about.
Anyway, no, not really odd at all. Not all wounds need by physical, you know. Mental fatigue and stress from dangerous attacks and environment can be just as debilitating as broken bones and severed muscles. Being fully combat effective (i.e. full HP) is just as much about mental and morale healthiness as it is about physical wholeness of body.
If some HP lose is due to wounds and some is due to moral loss - how do you determine which was the cause of each hit point?
Why do you fee the need to bother differentiating? The abstract nature of HP is precisely so you
don't need to determine what source of hit points are lost, just that some are lost.
The concept of hit points is a vast abstract designed to measure several factors. Each PC will likely have a different blend of those factors. A rogue's HP might be mostly luck and skill, a fighter's might be mostly physical resilience and gritty determination to keep soldiering on, a wizard's could be luck and mental acuity giving him the edge to move before solid blows can land, a cleric's hit points might be mostly granted by the favor of his god, whose divine providence makes an attacker's feet slip just enough to have a lethal attack miss by mere inches.
Cure Wounds (at least in the description in 5e) offers no text description as to the narrative of
how it restores hit points in the text description of the spell, it simply does. The narrative is left up to the DM. Maybe the cure wounds relieves mental stress, maybe it heals cuts and gashes, maybe it's divine magic (it is magic, after all) brings more "good luck" or divine favor in a blessing to the recipient? Any of those things are possible and left up to the player and/ or DM to determine along with the story of the game.
The very abstract nature of HP makes them fill all roles that are needed or possible, and there really isn't any problems with that as I see it anyway. If you want more differentiation, HP won't work for you as it seems not to (judging from your comments, anyway).