A troll's severed body parts remain 'connected' to the troll. Severed parts die when the troll regenerates them.
I am drawing my information from the 'Variant: Loathsome Limbs' sidebar under the Troll's entry in the 5th edition Monster Manual (page 291). This variant trait describes in mechanical detail what happens when you sever the arms, legs or head of a troll.
If the troll loses a body part, that body part can remain alive for a few hours apart from the rest of the body (specifically, until the troll finishes a short or long rest). After that, the troll regenerates the lost part. At that point, the severed part dies. No new trolls spawn from the severed body parts.
Additionally, the severed body parts remain part of the same consciousness. A severed arm has disadvantage on attack rolls unless the troll can see what the arm is doing. If the troll loses its head, the troll's body is blinded unless the head can see it. The severed head dies if the troll's body dies. These indicate that the severed body parts maintain some kind of mental connection to the original body. While biologically this might seem implausible, in D&D cosmology living creatures have souls. These severed parts would all share the same soul. This reinforces that severed parts would not form new trolls.
As for which part of the body is considered the origin, it would appear that the chest is always the part which regrows. You can cut off a troll's head, arms and legs, leaving only its chest. The head, arms and legs will regrow from the chest, with the old head, arms and legs dying.
But what of a bifurcated troll? That's unclear.
We come to your trickiest question: What happens if you surgically bifurcate a troll, creating two equal halves? The rules and Monster Manual provide no concrete guidance on this, but I can present some speculation.
Forming perfectly equal halves may not be possible. Human physiology has organs asymmetrically distributed inside the chest cavity. Simply cutting the troll in half could leave a critical organ in only one half of the body, and that half would be the one which regenerates. You might be able to work around this by making sure you cut that particular organ in half. But, having stepped well beyond the bounds of dealing hit point damage, it is possible that this might kill the troll irrespective of its Regeneration trait. But let's assume for a moment that the troll can survive this procedure.
Looking at the troll's behaviour when limbs are severed, we might assume that both halves of a bifurcated troll remain part of the same troll, sharing a soul and consciousness. By this assumption, only one half should survive. You can determine which half randomly.
However, the Monster Manual does say that "their regenerative capabilities make trolls especially susceptible to mutation." Surgically bifurcating a troll probably qualifies as an event likely to induce mutations. In such a scenario, I think creating two fully grown trolls is a reasonable mutation. The twins would almost certainly share their memories from when they were a single troll.
It is unclear whether an ongoing connection would exist between the twins and what the nature of that connection would be. They might continue to share a consciousness. They might have some twin-like pseudo-telepathic connection. Or they might be completely independent. There's probably some crazy wizard in your world (possibly yourself) investigating that question.