FLC OF DC9-30 OVERSHOT ASSIGNED ALT AFTER SOME CONFUSION AS TO THE ORIGINAL ALT CLRNC BY ZOB DEPARTING DTW.

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FO WAS FLYING AND I (PNF) WAS WORKING THE RADIOS. AFTER TKOF AND DURING CLBOUT; WE WERE ON DTW DEP CTL FREQ 132.02 AND WERE ON AN ASSIGNED HDG OF 210 DEGS MAGNETIC AND CLRED TO CLB TO 13000 FT MSL. CLBING THROUGH APPROX 11000 FT; DTW DEP SAID 'ACR XXX; CLB TO FL230 AND CONTACT ZOB ON 119.95.' I READ THE INSTRUCTIONS BACK AND CHANGED TO FREQ 119.95. ON INITIAL CONTACT; I SAID 'ZOB; ACR XXX IS WITH YOU; CLBING THROUGH 12500 FT FOR FL230.' ZOB SAID 'ACR XXX; YOU HAVE TFC AT X O'CLOCK LEVEL AT 14000 FT.' THE FO AND I AGREED TO START TO LEVEL OFF AND VERIFY THE CLB INSTRUCTIONS. I THEN ASKED 'ZOB; VERIFY XXX IS CLRED TO FL230.' ZOB SAID 'NEGATIVE ACR XXX; MAINTAIN 13000 FT.' ALTHOUGH WE HAD STARTED TO LEVEL OFF; DUE TO OUR CLB RATE APPROX 2500 FPM; WE OVERSHOT AND ASCENDED TO 13400 FT AND IMMEDIATELY DSNDED BACK DOWN TO 13000 FT. ZOB ASKED US TO 'VERIFY LEVEL AT 13000 FT.' WE DID. WE THEN HAD A DISCUSSION WITH ZOB AND TOLD THEM WE THOUGHT WE HAD BEEN CLRED TO FL230 BY DTW DEP ON 132.02. AFTER THIS DISCUSSION; THE FO AND I BOTH REALIZED WE MUST HAVE MISUNDERSTOOD DTW DEP AS DEP CTL DOES NOT CTL THE AIRSPACE UP TO FL230. LATER IN THE FLT; ZID TOLD US WE WERE TO CALL DTW DEP UPON ARR IN MSY. UPON ARR IN MSY; I CALLED DTW DEP AND TALKED WITH CTLR. THE CTLR SAID THERE WAS NOTHING ON THE COMPUTER OR LOG FOR ACR XXX. HOWEVER; I DID REVIEW THE INCIDENT WITH HIM. ALTHOUGH WE DID NOT LISTEN TO ANY TAPES; THE CTLR SAID I PROBABLY MISUNDERSTOOD THE CTLR AND THE INSTRUCTIONS WERE PROBABLY TO 'TURN TO A 230 DEG HDG' VERSUS 'CLB TO FL230' AS THEY DO NOT OWN AIRSPACE ABOVE 13000 FT. WITHOUT LISTENING TO TAPES AND FROM ZOB'S REACTION; I MUST ASSUME I MISUNDERSTOOD THE INSTRUCTIONS.