GND TFC CONFLICT AND DIFFERING PERCEPTION OF RIGHT-OF-WAY DURING TAXI AT MIA; FL.

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LANDED RWY 30; PARKED AT GATE XX; TOLD BY GND TO TAXI ON TXWY Q TO GATE. AS WE APCHED TXWY T TO TURN INTO THE CONCOURSE; WE HAD TO CROSS TXWY P. AN ACR B767 WAS ON TXWY P AND APPEARED STOPPED. WE PROCEEDED TO PASS IN FRONT OF HIM ONTO TXWY T TO THE GATE. GND CTL DID NOT TELL US TO DO ANYTHING ELSE AFTER OUR INITIAL CONVERSATION. AS WE PASSED IN FRONT OF ACR Y; HE TURNED ON HIS TAXI LIGHTS AND LNDG LIGHTS AND STARTED TO MOVE. IT WAS NOT A CLOSE ENCOUNTER; BUT ACR Y BELIEVED WE HAD CUT IN FRONT OF HIM AND WE; OF COURSE; BELIEVED ACR Y WAS AIMING FOR US. GND CTL DID NOT SAY ANYTHING TO EITHER OF US. WE PROCEEDED IN TO OUR GATE. I OVERHEARD A CONVERSATION BTWN ACR Y AND GND CTL. ACR Y CLAIMED WE CUT HIM OFF AS HE WAS TAXIING TO EITHER RWY 30 OR RWY 27L. GND TOLD ACR Y WE WERE SUPPOSED TO BE ON TXWY P AND HE DIDN'T KNOW WHAT HAD HAPPENED. MY CAPT CALLED THE TWR AND SPOKE TO THE SUPVR AND ASKED HIM IF THERE WAS A PROB. HE SAID NO. THE GND CTLR WAS SITTING NEXT TO HIM AND DID NOT SAY ANYTHING TO HIM ABOUT THE SIT. ACR Y AND OUR ACFT WERE NOT THE ONLY 2 PLANES IN THAT AREA. THERE WAS A LINE OF ACFT TAXIING FOR TKOF TO EITHER RWY 30 OR RWY 27L. THERE WERE AIRPLANES BEHIND ACR Y AND MORE ON TXWY P IN FRONT OF ACR Y. IT APPEARED TO US ACR Y WAS HOLDING SHORT OF TXWY T; SO WE COULD GO IN TO THE GATE. ACR Y DID NOT SEE IT THIS WAY. I FIND MIA TAXI MARKINGS VERY CONFUSING AND NOT WELL MARKED. IN THIS CASE; GND CTL SHOULD HAVE BEEN MORE SPECIFIC WITH US AND ACR Y. WE BOTH THOUGHT WE HAD THE RIGHT-OF-WAY.