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Firstly, Paul was not "reigning over Peter (by correcting him)".And why would Paul be interested in reigning over Peter (by correcting him) when he wasn't one of the twelve who were to reign over the 12 tribes?
Secondly, it was because Peter was in ANTIOCH, which was Paul's "home court".
Acts 11:26 (AKJV/PCE)
(11:26) And when he had found him, he brought him unto Antioch. And it came to pass, that a whole year they assembled themselves with the church, and taught much people. And the disciples were called Christians first in Antioch.
The disciples in Jerusalem were called Jews.