I thought that Satoshi gave the github keys to Gavin ?
No. Satoshi did not even have the project on Github originally. Gavin took over the project after Satoshi disappeared since he was the most experienced and active contributor at the time. I have found no evidence for Satoshi handing anything over except for Gavin's own statements. There is no indication from Satoshi that there was any plan for a "line of succession".
It is true that an implementation is a precise definition, but is such a precise definition necessary? You write as if Bitcoin does not have a mechanism for resolving disputes between nodes, but it does of course.
A precise definition of the consensus rules are necessary. Everyone must be following the exact same consensus rules, with its bugs, intricacies, and undefined behavior, in order to not have any accidental hard forks. The 2013 accidental hard fork is an example of such failure of different consensus implementations. On paper, all nodes were following the same rules. But in implementation, one version of the implementation had a bug which in turn caused an accidental chain fork. If everyone follows precisely the same rules, then this cannot happen. The only way for everyone to follow precisely the same rules is for every node to be share the same consensus implementation.