A B737-300 BEGINS TO TAXI AFTER RECEIVING DEICE INFO; BUT FINDS THAT DEICING IS NOT COMPLETE. COLLISION WITH DEICE TRUCK DAMAGES STABILIZER.

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WE PUSHED THE GATE IN ZZZ ON TIME AND WERE CLRED TO DEICE PAD #1. WE WERE SWITCHED TO DEICE FREQ AND SUBSEQUENTLY CLRED TO THE DEICE PAD. ONCE CLRED INTO THE FORWARD DEICE POS (2 POS ON EACH BAY) WE WERE THEN SWITCHED TO ICEMAN FREQ AND WE VERIFIED CONFIGN SET FOR DEICING AND PARKING BRAKE SET. DEICING BEGAN AT X TIME. WE WATCHED THE DEICING PROGRESS TO AFT OF THE WINGS AND LOST SIGHT OF THE TRUCKS AT THAT TIME. DEICE USES 2 TRUCKS; ONE ON EACH SIDE OF THE AIRPLANE. AFTER LOSING SIGHT FOR ABOUT 10 MINS; WE HEARD A DEICING COMPLETE RPT STATING 'FLT NUMBER; DEICING COMPLETE; TYPE 4; CONCENTRATION 100%; INITIALS XX.' FO READ THE RPT BACK EXACTLY; AND VERIFIED USING FULL CALL SIGN COMPANY NUMBER. ICEMAN THEN SWITCHED US BACK TO DEICE FREQ FOR TAXI INSTRUCTIONS OFF THE BAY. WE SWITCHED BACK TO DEICE AND RAN THE POST DEICE CHKLIST. I RECONFIGURED THE STABILIZER TRIM AND CYCLED THE CTLS 4 TIMES. WE THEN CALLED DEICE FOR TAXI. AFTER FO MADE THE CALL; I RELEASED THE PARKING BRAKE AND ALLOWED THE ACFT TO BEGIN TO ROLL FORWARD WITHOUT APPLYING PWR AS I WAS CONCERNED ABOUT THE POTENTIAL FOR PERSONNEL OR EQUIP BEHIND THE AIRPLANE. AS WE STARTED TO ROLL; DEICE CTL CAME BACK ON RADIO SAYING HE DIDN'T SHOW US AS BEING COMPLETE WITH DEICING. I IMMEDIATELY APPLIED THE BRAKES AND SET THE PARKING BRAKE. AS I BROUGHT THE AIRPLANE TO A STOP I HEARD ON THE RADIO; 'STOP; STOP; THERE ARE TRUCKS ALL AROUND YOU.' WE HAD ROLLED FORWARD APPROX 3-5 FT AND MADE CONTACT WITH THE DEICE BUCKET ON THE L LEADING EDGE HORIZ STABILIZER APPROX 18 INCHES FROM THE TIP. THIS RESULTED IN 2 2-INCH DENTS ABOUT 1/2 INCH DEEP ON THE LEADING EDGE. AFTER GETTING A TUG TO PUSH US AWAY FROM THE TRUCK; WE TAXIED THE AIRPLANE BACK TO THE GATE AND CALLED DISPATCH; MAINT CTL AND THE CHIEF PLT ON DUTY. THE AIRPLANE WAS TAKEN OTS AND WE WERE DEADHEADED TO ANOTHER CITY FOR THE NIGHT. DEICE OPERATES 9 BAYS IN DEICE PAD #1. EACH BAY HAS 2 AIRPLANES ON ITS LINE AT A TIME DURING PEAK TIMES. THIS WAS A PEAK TIME AND ALL BAYS WERE OCCUPIED WITH 2 AIRPLANES EACH SO THERE IS LOTS OF RADIO CHATTER. ICEMAN ALSO OPERATES ON 1 FREQ (DIFFERENT THAN DEICE) TO DEICE ALL 9 BAYS. WITH EACH BAY DEICING SIMULTANEOUSLY AND WITH ONLY 1 FREQ RPTING COMPLETED DEICING; THERE IS AMPLE OPPORTUNITY FOR CONFUSION OF CLRNCS. THIS COUPLED WITH THE TENDENCY FOR ICEMAN TO REDUCE CALL SIGNS TO NUMBERS ONLY CAN LEAD TO FURTHER CONFUSION. WE THOUGHT WE RECEIVED A CLRNC. IN FACT; THERE WAS NO QUESTION BTWN MYSELF AND THE FO AS TO WHETHER THE CLRNC WAS FOR US. WE USED OUR FULL CALL SIGN. WE WERE CLRED BY ICEMAN BACK TO DEICE. NOBODY ELSE RESPONDED TO THE CLRNCS OTHER THAN US. WE TOOK A COUPLE MINS TO RUN THE POST DEICE CHKLIST AND RECONFIGURE THE AIRPLANE AND CYCLE THE CTLS. NOBODY CAUGHT THAT THE WRONG AIRPLANE RECEIVED THE END OF DEICE RPT. WE THOUGHT THE AREA WAS CLR AND WE WERE ABOUT TO GET A TAXI CLRNC. THE ONE THING I SHOULD NOT HAVE DONE IS TO RELEASE THE PARKING BRAKE AND LET THE AIRPLANE ROLL UNTIL DEICE HAD ISSUED THE TAXI INSTRUCTIONS. IN THE FUTURE; I WILL ALWAYS RE-VERIFY CLRNCS WHENEVER THERE IS A CHANCE THERE ARE VEHICLES AROUND THE AIRPLANE. THERE IS LITTLE CHANCE THAT ICEMAN IS VERIFYING BY CALL SIGN READBACK THAT THE CORRECT ACFT IS RECEIVING THE CORRECT ALL CLR RPT. THIS IS UNLIKE ATC WHICH ACTUALLY ATTEMPTS TO VERIFY THAT THE RIGHT ACFT ACCEPTS THE CORRECT CLRNC. THE ICEMAN AND DEICE SETUP ON VHF RADIO FREQ; INSTEAD OF AIRPLANE INTERPHONE; GIVES YOU THE PERCEPTION OF AN ATC ENVIRONMENT. THIS IS A FALSE PERCEPTION.