You two are talking past each other. Because you are both right.

Let's recap what we actually know about Regeneration: Regeneration in Timelords is the result of a mutation. This mutation either occurred naturally through prolonged exposure to the time vortex (see A Good Man Goes To War) or was the result of genetic engineering, either from Rassilon himself (see Big Finish: Day Of The Master) or Techteun (see The Timeless Children). It gives the Timelords an internal storage of so called 'Regeneration Energy' (see The Magicians Apprentice / The Witches Familiar) that's used up during a regeneration. Each Timelord has only a limited amount of this energy and can't produce new ones. If not enough energy is left to make it to a new body, they die.

Those rules apply also to the Doctor, because they have undergone a Chameleon Arch transformation, making him, biologically speaking, into a Timelord.

However, the chameleon arch is a solution to a problem Chibnal himself introduced. A retcon you might say. So how might the regeneration limit have been solved if he had never become showrunner?

Well, we don't actually know how much Regeneration Energy a Timelord is capable of storing. We know how much energy they hold, enough for 12 regenerations. But that doesn't mean that's all they can hold.

Which brings us to The Time Of The Doctor. The Doctor is gifted new Regeneration Energy. We don't know how much. It might be enough for 12 regenerations it might be 1200.

The limit technically still exits, but the vagueness allows future writers to kick this can down the road indefinitely.