A B767 FLT CREW RPTED A HEARBACK READBACK ISSUE WITH ZMA; AND BTWN EACH OTHER; WHICH CAUSED A TRACK DEV.

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AFTER STARTING OUR DSCNT INTO MIAMI; WE WERE TOLD BY ATC TO TURN L 20 DEGS. I ACKNOWLEDGED THE CLRNC AND REPEATED 20 DEGS L. AT ABOUT THE SAME TIME WE WERE STARTING TO REVIEW THE NEW APCH PLATE FOR RWY 8 IN MIA AND AFTER I READ BACK THE CLRNC TO TURN 20 DEGS L; I WENT BACK TO REVIEWING THE APCH PLATE. THE PF MISTAKENLY TURNED THE HDG KNOB 20 DEGS R AND I DIDN'T CATCH HIS MISTAKE BECAUSE I WAS HEADS DOWN. AT ABOUT FL340 IN THE DSCNT; ATC SAID 'ACR XYZ; I SAID 20 DEGS L; NOT 20 DEGS R. LEVEL OFF AT FL330 AND TURN L NOW TO 180 DEGS.' WE COMPLIED WITH THE NEW CLRNC IMMEDIATELY BUT ZMA HAD TO LEVEL A CLBING ACR JET AT FL320 TO AVOID A POTENTIAL CONFLICT. WE SAW THE OTHER JET PASS ABOUT 6 MI OFF TO OUR N AND W AS WE WERE TURNING TO A SOUTHERLY HDG. THE FLT THEN CONTINUED NORMALLY TO AN UNEVENTFUL CONCLUSION. IN RETROSPECT; I SHOULD HAVE DOUBLECHKED THAT THE CLRNC WAS COMPLIED WITH BEFORE I GOT DISTR BY THE REVIEW OF THE NEW APCH PLATE. SUPPLEMENTAL INFO FROM ACN 592910: WE WERE REVIEWING NOTAMS AND NEW PROCS FOR LNDG ON THE NEW RWY 8/26 AT MIA. IN THE TURN; WE RECEIVED 1 TA ON TCASII ('TFC; TFC'). THE TCASII DISPLAYED AN ACFT 1000 FT BELOW PASSING L TO R 7-8 NM OFF THE NOSE OF OUR ACFT.