Explain how the First Doctor's regeneration changed his clothes, how K'Anpo had his next regeneration alive at the same time as him as a projection of his conscience, why the Third was able to postpone his death and regeneration for 3 weeks, why he couldn't regenerate without a "push" from K'Anpo, why Romana regenerated without dying, how she went through multiple forms (and sets of clothing), what the Fourth's "watcher" was, why the Sixth shrunk several inches _before_ dying and regenerating, how the Valeyard could exist if he took the Sixth's future regenerations, how the Sixth's future regenerations managed to kill him so the Seventh could be Time's Champion, why the Seventh's regeneration was delayed (but not prevented) by anesthesia, how the Time Lords were able to give the Master a new set of regenerations that had all his memories without even having his corpse, why the Tenth was able to redirect what should have been a regeneration to his hand and thereby grow a meta-Crisis Doctor and turn Donna half-Time Lord, why the Tenth was able to postpone his death and regeneration, why it set the TARDIS on fire when it finally happened, and why just about every regeneration seen on the show looked totally different from every other.
The answer you just gave to all those questions--that's how the Master was able to stop his regeneration.
All of those things aren't caused by "magic," as you say. They are just things you don't know. They could be caused by: the Master holding it in and not releasing it again, the Doctor having too much regenerative energy, so it changed his clothes, he absorb Cho Je so he can regenerate into him, the Third Doctor could hold it in just as the Tenth Doctor could, K'Anpo just sped up the regeneration, Romana's body was damaged by the Key to Time, she was a Newblood Time Lord so she had control of her regenerations more, an impression of the Fifth Doctor coming through before regeneration, the Sixth Doctor was regenerating while in the TARDIS, by generating a brand new timeline from that point, subconsciouly controlling his body to kill him, the chemicals delayed the trigger effect for regeneration, by constructing a new body for him and downloaded his consciousness, controllling his regeneration to heal his body then redirected it into the hand, Donna turning the hand gave her Time Lord DNA and the Meta-Crisis Doctor Human DNA, by holding it in so he doesn't regenerate, because holding it in for so long allowed more energy to build up, because the process is random and in flux, and the latest similar may have been engineered by the Time Lords for the Time War. Do these answer your questions?