RJ45 Cat5 Cat6
Cat 6 uses a different type to ensure the cables have separation all the way to the end to prevent near and far end cross tlak
Cat6a cables officially use the same rj45 connectors as a Cat6 (per standard), but the cat6 & cat6a cables are fine to use with cat5e connectors (speed not affected) in a pinch with no loss [only disadvantage could be crosstalk], (and as long as the cable/sheath fits the connector - ha). It's also, not ideal but, OK to use standard unshielded Cat6 connectors even if the cable is shielded and/or provides a ground wire (if not needed for the job).
CABLE = OFFICIAL CONNECTOR TO USE:
Cat5e Cable = cat5e rj45 connector
Cat6 Cable = cat6 rj45 connector
Cat6a Cable = cat6 rj45 connector
So I guess there is no Cat6a style connector, only just "Cat6" that works on all cat6 style cables. I also understand that sheath/cable gauge is not standard and may have to be manipulated, as you mentioned, if it's too thick to fit in cat6 rj45 connectors. Last but not least, I understand that cat5e connectors have a flat bottom whereas cat6 connectors stagger the cables only to prevent cross talk, due to the higher bandwidth.