is star-tipped peak once per turn per player?

Yes — Startipped Peak can only trigger once per turn for each player, though that's because of the scoring limit, not card wording.

Startipped Peak reads "When you hold here, you may channel 1 rune exhausted." A Hold score only happens during your own Beginning Phase when you maintain control, and Rule 470 caps scoring at any one battlefield to once per turn per player — Rule 471.2.c confirms Hold abilities "cannot be triggered more than once per turn for a player."

So each player gets at most one channel off it per turn, on their own turn. Leaving and re-taking the battlefield, or holding it again later, won't fire it a second time that turn.