FLT CREW OBSERVE A DEVELOPING CONFLICT WITH A CLBING MD80 AFTER ACKNOWLEDGING A ZKC CLRNC OF '310' INTERPED AS A HDG CHANGE; NOT A DSCNT CLRNC.

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WE WERE BEGINNING OUR DSCNT INTO WICHITA. WE WERE CLRED FROM FL390 TO FL350. WE THEN WERE CLRED TO FL330 WITH PLT'S DISCRETION TO FL240. WE ALSO WERE INSTRUCTED TO TURN L 10 DEGS TO A NEW HDG OF 260 DEGS. WE THEN RECEIVED A CLRNC TO '310.' WE READ BACK THE CLRNC AND TURNED TO A NEW HDG OF 310 DEGS. AFTER FLYING THIS HDG FOR ABOUT 2 MINS; WE SAW AN APCHING ACFT ALMOST SIMULTANEOUSLY ON THE TCASII AND VISUALLY. WE RECEIVED INSTRUCTION FROM ATC TO TURN TO 270 DEGS IMMEDIATELY AND WE WERE ALREADY IN THE TURN. THE OTHER ACFT PASSED BEHIND US ABOUT 4 MI AND CLBED THROUGH OUR ALT. ATC THEN ASKED US IF WE WERE DIRECT TO ICT TO WHICH WE REPLIED NO. HE INFORMED US WE HAD BEEN INSTRUCTED TO DSND TO FL310 AND WE HAD REPLIED. BOTH PLTS ON BOARD BELIEVED THE INSTRUCTION HAD BEEN A HDG CHANGE RATHER THAN AN ALT CHANGE. SUPPLEMENTAL INFO FROM ACN 621897: I WAS FLYING WITH AN INSTRUCTOR ON A TRAINING FLT. WE RECEIVED A CLRNC TO DSND TO FL240. I REPEATED THE CLRNC; AND ON THE ADVISE OF THE INSTRUCTOR REQUESTED TO STAY AS HIGH AS POSSIBLE; FOR AS LONG AS POSSIBLE TO CONSERVE FUEL. ATC GAVE US A CLRNC TO DSND TO FL330; PLT DISCRETION TO FL240; AS WELL AS INSTRUCTIONS TO 'FLY HDG 270 DEGS; THIS WILL BE VECTORS FOR THE DSCNT.' AS WE WERE DSNDING; THE NEXT CLRNC FROM ATC WAS '310.' SINCE WE WERE 'ON VECTORS FOR THE DSCNT;' WE TURNED TO A HDG OF 310 DEGS. A FEW MINS LATER; WE HAD A TCASII TA. IT WAS AN MD80. WE MAINTAINED VISUAL SEPARATION. AT THE SAME TIME; ATC INSTRUCTED US TO TURN TO HDG OF 270 DEGS AND DSND IMMEDIATELY TO FL310. LATER; THE ATC CTLR ASKED WHICH DIRECTION WE WERE FLYING AND WE ANSWERED HDG 310; WHICH WE REPEATED TO HIM. HIS RESPONSE WAS: 'I NEVER GAVE YOU THAT HDG.' BOTH THE INSTRUCTOR AND I AGREED THAT IT WAS THE HDG HE GAVE US AND WE CONVEYED THAT TO HIM. THE INSTRUCTOR TOLD ME THAT HE CALLED THE CTR AND THAT THEY HAVE REPLACED THE CTLR. FEW HRS LATER; THE CTR NOTIFIED THE INSTRUCTOR THAT THEY BELIEVED THAT THEY NEVER GAVE US A HDG. SINCE IT TRIGGERED AN ALARM AT THE CTR; THEY WILL PROCESS THE PAPERWORK. PERHAPS IF THE ATC CTLR WOULD HAVE USED THE CORRECT TERMINOLOGY AND INSTRUCTED US TO 'DSND TO; AND MAINTAIN FL310;' WE WOULD HAVE NOT INTERPED THE CLRNC AS A HDG CHANGE.