A C650 CREW; DEPARTING CGF; MISINTERPED A CLRNC; STARTING A CLB TO FL180 INSTEAD OF TURNING TO A HDG OF 180 DEGS.

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WHILE DEPARTING CUYAHOGA COUNTY ARPT (CLEVELAND); AN ALTDEV OCCURRED. OUR INITIAL CLRNC WAS; 'DIRECT ACO VOR; CLB TO 3000 FT.' UPON INITIAL CONTACT WITH ZOB; AN AMENDMENT WAS RECEIVED (AT THIS TIME THERE WAS 'XTALK' ON THE RADIO AND IN THE FLT DECK) AND UNDERSTOOD AS TO 'CLB TO FL180.' THE R SEAT CAPT REPLIED TO CLEVELAND; 'ROGER; ACFT X; CLB TO 180 DEGS.' HE THEN PLACED FL180 IN THE ALT ALERTER AND I RESPONDED 'CLBING FL180.' PASSING 4500 FT CLEVELAND INQUIRED (STRANGELY ACTUALLY) 'ACFT X; WHERE ARE YOU GOING?' OUR RESPONSE; 'DIRECT ACO.' 'NO; NO;' HE SAID; 'SAY YOUR ALT.' WE RESPONDED AGAIN; 'CLBING TO FL180.' HE ASKED; 'WHO GAVE YOU THAT?' WE RESPONDED; 'YOU DID.' WITH THIS CONFUSION; WE STOPPED THE CLB AT 4700 FT AND MAINTAINED THIS FOR A MOMENT UNTIL THIS LACK OF COM WAS SORTED OUT. (APPARENTLY THE AMENDMENT WAS NOT TO 'CLB FL180;' BUT WAS TO 'HDG 180 DEGS'). A FREQ CHANGE WAS GIVEN. A NEW AMENDED CLB AND HDG WAS RECEIVED; REPLIED TO AND COMPLIED WITH. SUPPLEMENTAL INFO FROM ACN 538899: WE STOPPED CLBING AND TURNED TO 180 DEGS AND STARTED TO DSND TO 3000 FT. ATC TOLD US TO LEVEL AT 4000 FT THEN CHANGED IT TO 6000 FT. WE PROCEEDED WITHOUT INCIDENT THEREOF. I FELT THERE WAS A DUAL MISCOM BTWN BOTH ATC AND THE PLTS. PLUS NONSTANDARD PHRASEOLOGY OF BOTH SIDES.