B737 CREW AT MIA TAXIED TO RWY 9L; THEN CHANGED TO RWY 12; AND AFTER TKOF; TURNED L AND CONFLICTED WITH TFC OFF RWY 9L.

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LOCATION -- 5 MI E OF MIAMI ARPT; WE WERE IN FLT. ABC CLRED MIAMI 7 WINCO TRANSITION DEP CTL FREQ LISTED ON THE DEP WAS 119.45 FOR OUR SECTOR. WE WERE CLRED TO TAXI TO RWY 9L. WHEN WE GOT THERE; THE NEXT 3 AIRPLANES TO TAKE OFF WERE ABD; ABE; AND US; ABC. WE WERE THEN TOLD TO TRANSITION TO RWY 12. WE DID SET UP PROPERLY; CLRED FOR TKOF; TOLD TO CONTACT DEP. NO FREQ WAS GIVEN. WE CONTACTED DEP PASSING 1000 FT; DEP SAID ACR X ABC TURN L HDG 020 DEGS; CLB AND MAINTAIN 7000 FT. WE ACKNOWLEDGED WITH FULL CALL SIGN AND COMPLIED. ABOUT THE TIME WE ROLLED OUT ON RWY 20; WE RECEIVED A TA ON TCASII; AND VISUALLY ACQUIRED THE ACR Y ACFT 300 FT BELOW; AND WE WERE GOING TO PASS SLIGHTLY BEHIND. IN A MOMENT; DEP CALLED US AND SAID -- 'ACR X ABC; I THOUGHT I WAS TALKING TO ACR X ABE -- YOU SHOULD HAVE BEEN GIVEN A TURN TO THE S AND HANDED OFF TO DEP ON 125.5.' WE FEEL WE WERE VIGILANT; COMPLIED WITH AND ACKNOWLEDGED ALL INSTRUCTIONS PROPERLY. THE TURN TO RWY 20 SEEMED REASONABLE AS WE WERE HEADED SE; AND OUR FIX WAS NW. I FEEL THIS INCIDENT OCCURRED; BECAUSE OF ALL THE SIMILAR CALL SIGNS COMING IN AND OUT OF MIA. THE FO AND I BOTH COMMENTED ON THE PROB AND TO BE EXTRA VIGILANT. AS A SIDE ACFT RIGHT AHEAD OF US WAS AN ACR X AFG; WE NEED TO SHUFFLE THE CALL SIGNS IN MIA SO THEY ALL AREN'T SIMILAR.