At the end of 2 Spencer is forcibly separated from Pinhead when he recollects his humanity at the behest of Kirsty as she escapes from the Doctor in Hell. Embracing his humanity, he fights the Doctor and loses in short order as an unseen result of the battle he is trapped in a torture column and Spencer's soul is unaccounted for.
In 3 Pinhead that is released from the column is entirely free of Spencer's influence. He's a rogue agent. It's an interesting bit of cosmology for the series, as it suggests that Cenobites are created by some sort of human or entity bonding, and in this case the Pinhead entity has become unbonded from Spencer and now exists without any of his human sense of order or honour. Spencer was a military man, and Pinhead was very regimented in Hellraiser I and II.
It's arguably less about the rules of Hell and more about Spencer's influence as without it Pinhead is a loose cannon but also kind of a doofus, making knock-off Cenobites, running around shouting and blowing things up, invading dreams, etc.
Pinhead throughout the series seems to have a soft spot for Kristy, willing to make deals instead of back tracking on them and seems to lose his cold nature as he has in one.