A B757 FLT CREW DEPARTING JFK ON THE JFK 9 DEP HAD THE WRONG CLB TRANSITION ENTERED IN THE FMS.

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TSTMS DELAYED OUR DEP BY 5 HRS. OUR ORIGINAL CLRNC WAS JFK 9 DEP; FLY RWY HDG. WHEN CONTACTING GND CTL; WE HAD A RE-FILE. AFTER RE-FILING OUR FLT PLAN; GND SAID TO MONITOR CLRNC DELIVERY FOR NEW RTE. JFK.9 DEP WITH BREEZY POINT CLB WAS ISSUED. I PUT THE BREEZY POINT CLB IN THE FMS. 30 MINS LATER; GND CTL SAID TO MONITOR CLRNC FOR ANOTHER RTE CHANGE. SUBSEQUENTLY GND GAVE SOME TAXI INSTRUCTIONS WHILE CLRNC GAVE RTE INSTRUCTIONS TO THE CAPT. I ASKED IF EVERYTHING ON THE CLRNC WAS THE SAME AS WHEN WE RE-FILED. CAPT SAID YES; EXCEPT FURTHER DOWN THE LINE (AWAY FROM NEW YORK). 15-20 MINS LATER; WE WERE CLRED FOR TKOF. CAPT VERIFIED CANARSIE. CTLR REPEATED CANARSIE. IN MY MIND; CANARSIE MEANT (NOT RWY HDG); STILL CLRED KEN.9 (JFK.9) DEP AND BREEZY POINT CLB. THE CTLR MEANT CANARSIE CLB; NOT DIRECT CANARSIE VOR. JFK.9 GOES DIRECT CRI VOR THEN DIFFERENT CLBS ARE AVAILABLE. I STILL BELIEVED THE BREEZY POINT CLB WAS OUR CLRNC. FOR THE CAPT SAID NOTHING CHANGED. ACTUALLY THE CANARSIE CLB WAS THE CLRNC AND THE CAPT THOUGHT THAT IS WHAT WAS IN THE FMS (ACTUALLY BREEZY POINT WAS IN). OVER THE CRI VOR; A VECTOR TO 090 DEG HDG WAS GIVEN (WHICH IS STANDARD OCCURRENCE). THE CTLR THEN ASKED WHY WE DIDN'T TURN ON THE CANARSIE CLB. WE WERE JUST PAST THE CRI VOR WHEN HIS VECTOR TO 090 DEGS WAS GIVEN; THEREFORE; I DON'T BELIEVE ANY AIRSPACE VIOLATION OCCURRED; BUT SINCE THE FMS USUALLY LEADS THE VOR; THE CTLR COULD SEE SOMETHING WASN'T QUITE THE SAME AS ALL THE OTHER ACFT. WE SAID WE WOULD CHK OUR INPUTS TO THE FMS TO SEE WHAT THE DIFFERENCE MIGHT BE. AGAIN; I BELIEVE MOST ACFT WERE GETTING A VECTOR OFF THE DEP NEAR THE CRI VOR; AS DID WE; THEREFORE; WE HAD YET TO FLY ANY CLB PORTION OF THE DEP. THE CTLR MUST HAVE SEEN WE DID NOT LEAD THE TURN AS THE OTHER FMS ACFT -- YET IN MY OPINION; WE WERE STILL WITHIN TOLERANCE. HAVING A CANARSIE CLB ALONG WITH OTHER CLBOUTS ALL STARTING AT THE CANARSIE VOR; LED TO SOME OF MY CONFUSION WHEN THE CAPT AND CTLR VERIFIED 'CANARSIE.'