tolkiens legendarium - Is Gandalf the Gray's power actually limited or did he merely promise to not use it?

We know that the Maiar sent to Middle-earth as Istari are not using their full Maiar power. Gandalf the Gray is more than a human power wise, but not nearly equal to his full real power.

Is it known (from canon) whether there is something explicitly restricting Olorin's power when he's Gandalf the Gray, or is he merely prohibited from using it but still has it at his disposal?

Question arose from this comment. Daniel was using Saruman having greater powers once he turned "bad" as his reasoning.

Answer

Tolkien makes a couple of explicit references to some "rules of engagement" (my term, not his):

it was a humbling and abnegation of himself in confirmity to 'the Rules'

And

He is still under the obligation of concealing his power and teaching rather than forcing or dominating wills

It is unclear whether these restrictions were "hardcoded", or merely the choice of the Istari to obey them. Given the Istari were given enough free will to get distracted (Radagast) or become corrupted (Saruman), I would extrapolate that these rules were not enforced. Rather they were instructed not to directly engage Sauron and his minions unless failure of their mission is the only other option, to conceal their power and to exploit the "soft" powers to allow the races of Middle Earth to defeat Sauron through their own means, not by direct conflict between Maiar.

In addition to those his "mortal" form also acted as part of these rules, albeit an unchangeable and hence "enforced" part. As Tolkien said,

they were embodied in physical bodies capable of pain, and weariness, and of afflicting the spirit with physical fear, and of being 'killed', though supported by the angelic spirit they might endure long, and only show slowly the wearing of care and labour.

Pain, weariness, fear and death were all hazards that the Istari would either not face at all without the restrictions of their physical forms, or would only face under much more limited circumstances. The concept of death definitely existed for them, as Gandalf really 'died', and was changed.

All quotes are from Letter 156.