As stated in the headline, I would like to eventually apply for a C Bewilligung but I have been getting some mixed information about whether or not moving across communes (Stadt or Gemeinde) has any effect on the process, restarting the counter of 5 years so to speak. I live in Kanton Zürich and the moving would be within the same Kanton. I read up the official government site and, to my understanding, it had no mention of it. Does anybody have any experience with something similar?
No, the gemeinde has no say in anything permit related, foreigners are a cantonal matter until they want to naturalise.
The counter for a C permit doesn't reset at all even when moving cantons (however different cantons do interprent the C permit regulations a bit differently, e.g. some would count L permits on a permanent contract and others wouldn't, in some you could apply alone and in others you have to apply as the whole family etc.).
But moving communes within a single canton is absolutely safe in this regard.
Ask your Gemeinde.