MISUNDERSTANDING OF THE APCH TO BE FLOWN AT SAN JOSE DEL CABO; MEXICO; BRINGS THE CTLR TO ASK 'WHAT APCH WAS FLOWN?'

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INBOUND TO SAN JOSE DEL CABO; MEXICO; RECEIVED RTE CHANGE FROM MZT CTR. AFTER LAP V1E MISEX SJD. CHANGED FREQ TO CABO APCH AND ISSUED ANOTHER RTE CHANGE TO V1W. REQUESTED TO REMAIN ON V1E AS WE WERE ESTABLISHED ON AIRWAY; OUTBOUND FROM LAP. APCH DID NOT ADVISE OF RWY IN USE OR APCH TO EXPECT. REQUESTED WINDS AND WX AND TOLD RWY 34 WAS ACTIVE RWY. CREW BRIEFED AND EXPECTED VOR DME 3 CIRCLE TO RWY 34. APCH CTLR CLRED US TO DSND TO 5000 FT AND CROSS THE 'ARPT' AT 5000 FT (NOT THE SJD VOR). OVER THE ARPT; APCH CLRED US FOR THE VOR DME-1 RWY 16/34. WE RPTED THE ARPT IN SIGHT; AND EXPECTED THE VISUAL AS APCH DID NOT ESTABLISH US OVER THE VOR FOR THE APCH; BUT RATHER HAD US CROSS THE 'ARPT.' AFTER PASSING ARPT; PF EXECUTED A TEARDROP ENTRY TO FINAL; TURNING INSIDE PUBLISHED PROC APPROX 3 MI. AT 3-4 DME ON FINAL; ON GLIDE PATH; AND STABILIZED FOR LNDG; APCH CTL ASKED IF WE HAD EXECUTED VOR DME-1 RWY 34; AND QUESTIONED THE PROC FOLLOWED. AFTER ARR AND SHUTDOWN; WE DISCOVERED THE DIFFERENCE BTWN THE VOR DME-1 RWY 34; VOR DME-1 RWY 16; AND VOR DME-3 RWY 16 CIRCLE TO RWY 34. ALTHOUGH WE DID NOT EXECUTE THE VOR DME-1 RWY 34 AS PUBLISHED; WE FLEW THE VOR DME 3 CIRCLE TO RWY 34 AND DID NOT LEAVE MDA UNTIL ESTABLISHED ON FINAL WITH ARPT IN SIGHT. CONFUSION WAS BROUGHT ABOUT BY DIFFERENT VOR DME APCHS AND LANGUAGE BARRIERS. IN RETROSPECT; I SHOULD HAVE QUESTIONED THE CTLR'S CLRNC AND EXPECTATIONS. WX AT ARR ARPT WAS CLR WITH LIGHT WINDS FROM THE N. VISIBILITY WAS UNRESTR.