CARJ FLC ASSUME TWR ADVISORY TO 'BEGIN L TURN' MEANT TO CLB ON COURSE; CAUSING CONFLICT WITH ANOTHER ACFT.

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OUR FLT WAS ISSUED A PDC THAT INCLUDED THE PALM SPRINGS 4 DEP PROC. THE PROC FOR DEPARTING RWY 13R WAS A VECTOR; 'HDG 110 DEGS AND EXPECT RADAR VECTORS TO ASSIGNED ROUTING OR FIX.' IN COMPLIANCE WITH OUR SOP; RWY HDG WAS MAINTAINED; '130 DEGS;' UNTIL 400 FT AGL BEFORE INITIATION OF ANY TURNS. AT 400 FT AGL AFTER TKOF; A L TURN TO HDG 110 DEGS WAS INITIATED. AT APPROX THE SAME TIME AS THE TURN WAS BEING COMPLETED; THE TWR CTLR STATED 'CONTACT DEP; START YOUR L TURN.' BECAUSE WE WERE FLYING OUR PREVIOUSLY ISSUED CLRNC; WE THOUGHT THE CTLR MEANT FOR US TO PROCEED ON COURSE. THIS WOULD HAVE BEEN A L TURN OF ABOUT 90 DEGS FROM RWY HDG. CONTACT WITH DEP CTL REVEALED THAT WE HAD TURNED TOWARD ANOTHER ACFT; WHICH THEN APPEARED ON TCASII. DEP CTLR THEN INSTRUCTED AN IMMEDIATE R TURN TO 110 DEGS. TCASII INFO AND VISUAL CONTACT WITH THE TFC MEANT THAT THE R TURN WOULD HAVE RESULTED IN DANGER; THUS THE CLRNC WAS NOT COMPLIED WITH. THE CTLR THEN VERIFIED OUR HDG AND ASKED US TO REMAIN PRESENT HDG. ALTHOUGH TA WAS RECEIVED ON TCASII; NO RA OCCURRED. PLT SHARES RESPONSIBILITY WITH ATC.