What essentialy constitutes a github api call. The documentation states that it has a limit on the number of api calls per hour. Does a call consist of any communication to the server like push or pull, or any git operation like commit?
Using any of the REST API endpoints listed in their API v3 documentation would count against the API v3 rate limit. There is also a GraphQL API; using any of those endpoints would also count as an API call against the API v4 rate limit.
It looks like the rate limits for the two APIs are unrelated: you could conceivably use both APIs to their respective limits in the same time span.
Regular git commands have their own protocols. They would not count against your API rate limit as they don't use the API.