I'm referring to the scene at the end of the first film where he is shot "dead" by Agent Smith, then after some scenes of his heart monitor flat-lining and Trinity telling him he can't be dead, his heart monitor shows a pulse again and he gets back up to continue fighting Agent Smith, with now far less effort involved on his part.
Agent Smith: Check him.
Agent Brown: He’s gone.
Agent Smith: Goodbye, Mr. Anderson.
Trinity: Neo, I’m not afraid anymore. The Oracle told me that I would fall in love, and that that man, the man who I loved would be The One. So you see, you can’t be dead. You can’t be, because I love you. You hear me? I love you…. Now get up.
Neo, The One: No.
Morpheus: He is The One.
What I want to know is how exactly does he "come back to life"?
We know that Neo was always the One, even before this scene, so it wasn't that he somehow needed to die and be reborn to become the One or anything like that. That said, this scene with Neo meeting the Oracle does still add some confusion, but I assumed she was just being flippant, not literally predicting his death and miraculous resurrection:
Oracle: Sorry, kid. You got the gift, but it looks like you’re waiting for something.
The Oracle: Your next life maybe, who knows? That’s the way these things go.
So did he literally die, or was he "nearly dead", dying, "as good as dead" but not actually dead yet? After all, flat-lining heart monitors don't mean actual death, since in real life people can be resuscitated and their heart rate returned to normal, and that doesn't mean they actually died, just that they nearly died.
What makes sense to me personally is that he was only dying (not dead), but Trinity's words helped him to realise, finally, at last, right at the last moment before he died, that whatever he experiences in the Matrix really isn't real. That he wasn't really dead or dying or anything; in reality, he was actually fine. Since "the mind makes it real", then once he'd realised this, his heart rate returned to normal and he got up, realising that he isn't dead because none of what's happening in the Matrix is real (him "seeing code" before he destroys Smith showing that he's now "seeing the Matrix for what it is"). This would also essentially "unlock his potential as the One", being able to defeat Smith with ease, stop bullets, fly, etc.
Whilst that makes sense to me (certainly more sense that the alternative), does the evidence in canon material support this, or does the evidence instead support that he did actually die and come back to life (thus proving my above interpretation as incorrect)?
If he really did die, then how exactly does he come back to life? The "power of love", the "magic of prophecy", what? In real life, people don't actually die from multiple bullets, and then miraculously come back to life, so there must be an in-universe explanation of how he could have achieved this magical feat of resurrection.
I'm only interested in in-universe explanations, ideally backed up by canon sources, not out-of-universe explanations about how he's an analogy for Jesus or anything like that.