B737 LNDG FLL TURNS TO 190 DEG HDG; ATC CONTENDS 190 WAS SPD ASSIGNMENT.

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ATC INSTRUCTIONS MISINTERPED; POSSIBLE TFC SEPARATION PROB. WE WERE ON 280 DEG ASSIGNED HDG; L DOWNWIND FOR RWY 9L FLL. AT 4000 FT MSL; WE WERE GIVEN WHAT WE BELIEVED TO BE A 190 DEG HDG; AND READ IT BACK AS SUCH; AND BEGAN THE L TURN; STANDARD RATE; LEVEL AT 4000 FT. THIS SEEMED LOGICAL AS WE WERE AT ABOUT THE NORMAL AREA FOR A BASE TURN. ABOUT HALFWAY THROUGH THE TURN; I VISUALLY ACQUIRED ANOTHER ACFT AT MY 2 O'CLOCK POS AND CHKING THE TCASII; I COULD SEE HE WAS 1000 FT ABOVE US AND DSNDING. HE WAS ON FINAL FOR RWY 9L AS WELL. FROM THE TCASII; AND MY VISUAL; IT APPEARED THAT A CONFLICT MIGHT RESULT IF WE CONTINUED THE TURN. SIMULTANEOUSLY; WE GOT TA FROM TCASII; AND THE CTLR ASKING US WHY WE WERE IN THE TURN. WE WERE ROLLING OUT OF THE TURN AT ABOUT 210 DEG HDG WHEN THE CTLR SAID 'I GAVE YOU 190 DEG SPD; NOT A 190 DEG HDG.' AT THIS TIME; HE DIRECTED US TO DSND TO 2000 FT AND FLY OUR CURRENT HDG. I HEARD HIM TELL THE OTHER ACFT TO SLOW HIS DSCNT. THE TA CEASED ON TCASII; AND WE HAD THE A320 VISUALLY AT ALL TIMES. HE TOLD THE CTLR THAT HE HAD US VISUALLY. THE CTLR CLRED THE A320 FOR THE VISUAL APCH TO RWY 9L AND SHORTLY AFTER; HE CLRED US FOR THE VISUAL APCH. TFC SEPARATION (5 MI) AND LNDG WERE UNEVENTFUL FROM THIS POINT. UPON SHUTTING DOWN AT THE GATE; I CONTACTED MIAMI APCH ON MY CELL PHONE AND EXPLAINED TO THE SUPVR WHAT I THOUGHT OUR CLRNC HAD BEEN AND WHAT HAD HAPPENED. I ALSO ASKED IF THERE HAD BEEN A SEPARATION PROB. HE SAID HE HAD REVIEWED THE SIT WITH THE CTLR AND DID NOT SEE WHERE WE HAD VIOLATED ANY MINIMUM SEPARATION DISTANCES; BUT COULD HAVE IF WE HAD NOT ROLLED OUT OR ACQUIRED THE OTHER TFC VISUALLY. HE SAID HE WOULD CONTACT THE OTHER PLT; AND IF HE HAD NO PROB; THEN APCH WOULD HAVE NO PROB EITHER. WE DISCUSSED THE EVENTS; RADIO CALLS LEADING TO THE POTENTIAL CONFLICT; AND WHAT WE COULD DO TO PREVENT SUCH EVENTS IN THE FUTURE. THE SUPVR WAS VERY UNDERSTANDING OF THE EVENTS AND AGREED THAT THE TURN MADE SENSE; AT THAT POINT; WHEN NORMALLY WE TURN BASE. WE TALKED ABOUT THE IMPORTANCE OF PROPER RADIO TERMINOLOGY AND READBACKS. WE THOUGHT WE READ BACK 190 DEG HDG; BUT BOTH MYSELF AND THE FO COULD NOT REMEMBER IF HE SAID DEGREE. THE SUPVR WAS ABLE TO SPEAK TO THE OTHER CAPT; AND HE SAID HE HAD NO PROB WITH ANY TFC SEPARATION AND THAT HE SAW US THE ENTIRE TIME. THE APCH SUPVR CALLED ME BACK AND SAID NO FURTHER ACTION WAS NEEDED AND THE EVENT WAS CLOSED WITH NO CONFLICT NOTED. THE LESSON LEARNED IS TO LISTEN UP ON THE RADIO; ASK FOR CLARIFICATION IF UNSURE; AND USE PROPER RADIO TERMINOLOGY. ALSO; NEVER ASSUME THAT BECAUSE THE TIMING AND LOCATION AND HDG OF THE TURN 'FEEL' RIGHT; THAT IT IS RIGHT. HAD THE EVENT TAKEN PLACE IN IMC; IT MIGHT NOT HAVE FELT AS COMFORTABLE WHEN ATC AND THE TCASII START BARKING AT US AT THE SAME TIME. MY THANKS TO THE CTLR AND THE OTHER CAPT FOR THEIR UNDERSTANDING OF THE SIT. WE ALSO CALLED OUR COMPANY ATC REPRESENTATIVE AND INFORMED HIM OF THE EVENTS.