PLT RPTS AN NMAC WITH A SEMINOLE AFTER DEPARTING PARALLEL RWYS WITH A CONFUSING ATC CLRNC.

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I WAS ON A LIFEGUARD FLT. ARPT WAS SEVERELY BACKED UP -- ABOUT 20 PLANES WAITING BTWN 2 PARALLEL RWYS. TWR GAVE ME PRIORITY DEP AS A LIFEGUARD FLT AS I HAD PATIENT ON BOARD. AFTER TKOF ON RWY 21 I WAS TOLD BY TWR 'START YOUR R TURN WBOUND AND PROCEED ON COURSE; THE SEMINOLE WILL OFFSET FOR YOU.' I WAS JUST OFF RWY 21L AND THE SEMINOLE WAS JUST OFF RWY 21R AND STAYING IN THAT PATTERN. I HAD ALSO HEARD THE CTLR TELL THE SEMINOLE WHO HAD JUST TAKEN OFF AND WAS A LITTLE BIT AHEAD OF ME TO MY R IN THE PATTERN FOR THE R RWY; TO OFFSET FOR TFC BEHIND HIM; A LIFEGUARD FLT NEEDING TO DEPART TO THE W. I STARTED MY R TURN TO 270 DEGS AND WOULD HAVE CROSSED BEHIND THE SEMINOLE. I SUSPECT THE CTLR MEANT TO TELL THE SEMINOLE TO 'EXTEND' FOR A LIFEGUARD FLT TURNING BEHIND HIM BUT HE SAID OFFSET. AFTER I MADE THE R TURN TO 270 DEGS; THE SEMINOLE MADE ESSENTIALLY A 'U' TURN AND WAS NOW HEADED RIGHT FOR ME. HE WOULD HAVE BEEN HDG ABOUT 360 DEGS. I WAS ABOVE HIM AT THIS POINT BUT HE WAS RAPIDLY CLBING. I HAD HIM IN SIGHT AT ALL TIMES AND WAS WAITING FOR HIM TO ARREST HIS ASCENT. IF NOT I WOULD HAVE STARTED EVASIVE MANEUVERS. IT WAS APPARENT HE STOPPED ASCENDING WITHIN ABOUT A HUNDRED FEET OF ME SO I CONTINUED ON COURSE. THE SEMINOLE THEN CALLED THE TWR AND SAID THEY HAD TO MAKE EVASIVE MANEUVERS TO AVOID HITTING AN AIRPLANE. THE CTLR CALLED ME AND ASKED IF I HAD SEEN THE SEMINOLE. I REPLIED I HAD MY EYE ON HIM THE ENTIRE TIME AND THOUGHT HE WAS 'OFFSETTING' FOR ME BECAUSE HE HAD INSTRUCTED ME TO START MY WBOUND TURN. THE CTLR REPLIED VERY QUICKLY 'I THOUGHT YOU WERE GOING TO TURN OUTSIDE OF HIM; FREQ CHANGE APPROVED GOOD DAY.' A RELATIVE OF MINE IS AN AIR TFC CTLR. I TOLD HIM THE ABOVE INFO AND HE SAID WHAT DOES OFFSET MEAN; IT'S NOT IN THE CTLR'S DICTIONARY. I HAVE HEARD THE WORD OFFSET USED MANY TIMES OVER THE LAST 20 YRS BUT WAS NOT AWARE IT WAS UNOFFICIAL. TO ME; WHEN THE CTLR TOLD THE SEMINOLE TO OFFSET FOR THE ACFT BEHIND HIM IT MEANT HE WAS TELLING THE SEMINOLE TO KEEP GOING STRAIGHT AND STAY S FOR THE ACFT TURNING WBOUND BEHIND HIM. IT WOULD APPEAR THE SEMINOLE PLT THOUGHT IT MEANT SOMETHING ELSE HENCE THERE WAS A CONFUSION AS TO WHAT THE WORD OFFSET MEANT WHEN THE CTLR USED IT. I DID NOT FEEL IN DANGER AT ANY TIME BUT DID FEEL AS THOUGH I WAS GOING TO HAVE TO USE EVASIVE MANEUVERS TO AVOID DANGER HAD THE SEMINOLE PLT NOT SEEN ME. WE WERE BOTH CLBING SO THE VERT CLOSURE RATE WAS NOT ALL THAT GREAT BUT THE HORIZ WAS.