Hit Points = Stress, Fatigue, and Pain tolerance you can take. Thus why Illusions "hurt" you, why Psychic Attacks can tear you down, why being Missed by an attack can still leave an effect, why Fear can be deadly.
0 Hit Points for PCs = Falling into Shock, likely the Hollywood version which seems to mix the medical version with the Combat Stress Reaction. The target is "unconscious", but more disconnected with reality, not caring or understanding why he should go on. Despair, Exhaustion, Physical Lack of Adrenaline to the point the body starts to slowly shut down (Ie saving throws), and likely will give up if taken too far (the -bloody hp range).
Thus why a Warlord's words can still reach the ally who is unconscious. To give them hope, a goal, to press on just a little more, to stand up and do one more task and then he can rest.
Never underestimate the power of the mind. Placebos, Adrenaline, or just simple Focus on a Goal.
Surges are more of what a PC can take Physically (& Mentally) in that day. Using a Surge is more like pushing past the barrier of shock just a little more, and feeling the pain of going past the limits. Onset of a headache, wounds that you put to the back of your mind, the trembling hands that you just simply ignore and press on. Meditation, relaxation, simply sitting back and taking an Ale Break are methods to push past the stress normally (Short rest), though you may be able to catch a breath in combat (Second Wind), while others have a prescence or a way with words that simply help you push past (Leader Words). Its why some PCs just can't recover later on, while everyone else can still charge forwards. They just can't ignore anymore of the injuries, stress, headaches, or the shaking limbs anymore then they have.
Thus for truly magical healing, is recovering from that stress without using a Surge (or recovers Surges themselves). Bards, Warlords, Clerics, Sentinels, and Ardents class features all aren't magical healing really. Its just truly inspiring or driven, you easily push past the shock. Shaman's though is, as another is healed without spending a Surge. Paladin's Lay on Hands don't so much as cure an Ally's wounds, but takes them for himself to burden so it is magical in one sense (when used on himself, more like a quick prayer/meditation of faith). Cleric's Utilities that heal without a surge, such as Cure Light Wounds utility IS magical. Cleric's Astral Seal is "magical" as it heals an ally without surges being lost.
Its also what separates the PCs from the normal monsters. They can push past their limits, where as a NPC usually cannot without magical help. Of course some NPCs are known to have Limits beyond what is the norm, such as Elite and Solos.
Thus its been in my mind, anything that recovers stress (aka HP) without the use of surges, or recovers Surges themselves, is more of the magical healing in previous editions.