It wasn't until later that they tried to give O'Brien a more traditional rank and came up with Senior Chief Specialist in "Shadowplay."
That would rather easily fit a consistent pattern. His rank is Chief (more specifically Senior Chief Petty Officer), Specialist (more specifically, his speciality is Ops). It's just that the current USN word for both the level of seniority and the line of work of an enlisted person is "rating" rather than "rank".
But "rank" is simpler, and it's partially in line with Royal Navy tradition, and both these arguments are good ones for Star Trek which wants to portray a simple to understand, improved future whilst suggesting the adventurous days of the sailing ship era.
O'Brien could thus be excused for saying the same thing in different ways: his rank is
Senior Chief (Petty Officer), Specialist (of Operations) ["Playing God"],
(Senior) Chief (Petty Officer), (Specialist) of Operations ["Tribunal"] or
(Senior) Chief Petty Officer, (Specialist of Operations) ["Hippocratic Oath"],
depending on how much of the mouthful he decided to swallow at a given instance.