C560 FLT CREW RECEIVES TCAS RA DURING INITIAL CLIMB DUE TO ATC CONFUSION REGARDING THEIR IFR/VFR STATUS.

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WE ARE A CITATION 650; FILED IFR FROM SUA TO ZZZ; RECEIVED CLRNC PRIOR TO TAXI. CLRED AS FILED; MAINTAIN 3000 FT; GIVEN A SQUAWK AND DEP FREQ. AT XA30 AND A CLR DAY; WE WERE CLRED BY TWR FOR TKOF FROM RWY 12. I THOUGHT IT WAS STRANGE WE WERE NOT GIVEN A HDG FOR AFTER DEP AND EVEN QUERIED THE OTHER PLT. HE WAS THE ONE WHO HAD RECEIVED THE CLRNC; HE SAID 'YUP; AS FILED.' SO ON THE TKOF ROLL I ASKED IF HE WAS GONNA MAKE A L OR R TURN TO PICK UP OUR COURSE FOR 'AS FILED.' HE SAID A L TURN AND WE DEPARTED RWY 12 CLBING TO 3000 FT. PASSING THROUGH 2500 FT; I HEARD TWR SAY 'CESSNA XXX; SQUAWK 1200.' I ASSUMED HE WAS TALKING TO THE WRONG PLANE AND SAID 'CESSNA XXX IS IFR TO CHICAGO.' TWR CAME BACK AND SAID 'I KNOW THAT; DSND AND MAINTAIN 1500 FT AND CIRCLE THE FIELD OVERHEAD.' NONE OF THIS IS COMPUTING FOR US; IT WAS VERY CONFUSING AND WE KEPT THINKING HE MEANT THE CALLS FOR SOMEONE ELSE; FINALLY DECIDING TO FOLLOW HIS DIRECTIONS WE LEFT 3000 FT FOR 1500 FT. AS WE DSNDED; THERE WAS AN ACFT CLBING UP BELOW US. WE RECEIVED AN RA AND BEGAN TO CLB TO AVOID THE TFC CLBING BELOW US. THE OTHER ACFT WAS RIGHT UNDER US AND NOT IN OUR SIGHT. IF WE HAD NOT HAD TCAS THAT DAY; WE QUITE LIKELY WOULD HAVE DSNDED DOWN INTO HIM; ON TWR'S DIRECTIONS. A FEW SECONDS LATER; TWR TELLS US TO MAINTAIN 3000 FT AND CALL DEP. ONCE WE WERE UNDERWAY; I CALLED BACK ON TWR FREQ AND ASKED WHAT THAT WAS ALL ABOUT. THE CTLR SAID HE THOUGHT WE WERE VFR WHEN HE CLRED US FOR TKOF AND WAS RUSHING TO KEEP US OUT OF CTR'S AIRSPACE AS HE CORRECTED HIS ERROR. I PHONED THE TWR SUPVR THE NEXT DAY AND TOLD HIM THAT WE WERE DSNDED INTO ANOTHER ACFT BY A CTLR. HE SAID BECAUSE THEY ARE 40 MI FROM THE RADAR SITE; THEY SOMETIMES DON'T PICK UP ALL THE TFC ON THEIR SCREEN. HE APOLOGIZED PROFUSELY. I DID MENTION THAT IF IT WEREN'T FOR TCAS WE QUITE LIKELY WOULD HAVE HAD A MIDAIR. HE SAID 'GOOD THING YOU ALL HAVE TCAS NOW.' I SHOULD HAVE QUERIED TWR ABOUT A HDG ON OUR TKOF CLRNC WHICH MIGHT HAVE INDICATED TO HIM WE NEEDED A RELEASE. AND; OF COURSE; HE SHOULD NOT HAVE CLRED US FOR TKOF WITHOUT ONE.