A C172RF THINKS THAT HE WAS CLRED TO CROSS RWY BUT ATC SAID THAT HE WASN'T.

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THE FOLLOWING EVENT OCCURRED WHILE REPOSITIONING; BY TAXI; FROM THE W SIDE TO THE S SIDE OF ISP ARPT. I INITIALLY CONTACTED LONG ISLAND TWR ASKING PERMISSION TO REPOS FROM THE W SIDE TO THE BASE OF THE TWR (THE S SIDE). THE CTLR REPLIED 'START TAXI VIA TXWY W UP TO BUT HOLD SHORT OF RWY 6.' I READ BACK THE INSTRUCTIONS STATING TO START TAXI VIA TXWY W HOLDING SHORT OF RWY 6. AS I WAS TAXIING; THERE WAS AN ACFT ON TXWY W HOLDING SHORT OF RWY 6; PERFORMING A RUN-UP. THE CTLR ASKED IF I WAS ABLE TO GET AROUND THE ACFT. I REPLIED THAT I WAS ABLE. THE CTLR THEN SAID 'USE CAUTION TAXIING AROUND THE ACFT AND CROSS RWY 6.' AS I WAS TAXIING ACROSS RWY 6; I NOTICED AN ACFT ON SHORT FINAL FOR RWY 6. I WAS CLR OF THE RWY BEFORE THE ACFT TOUCHED DOWN. THE CTLR THEN CAME ON THE FREQ AND SAID 'YOU WERE INSTRUCTED TO HOLD SHORT OF RWY 6.' I REPLIED 'YOU CLRED ME ACROSS RWY 6.' THE CTLR SAID 'CALL THE TWR WHEN YOU PARK.' I REPLIED 'ROGER; I WILL CALL YOU WHEN I PARK.' I CALLED AND TALKED TO THE CTLR A FEW MINS LATER AND HE SAID 'YOU WERE INSTRUCTED TO HOLD SHORT OF RWY 6.' AGAIN I TOLD HIM THAT HE HAD CLRED ME ACROSS THE RWY. I FEEL THAT PLTS AND CTLRS NEED TO LISTEN TO EACH OTHER AND DECIPHER WHAT IS SAID BEFORE ACTING ON IT.