ZOB Controller providing OJT described a deviation event when coordination with an adjacent sector was confused; reporter suggests changing the procedure because of the current multiple numbers of controllers involved.

Narrative

DKK Sector routinely works aircraft that never enters DKK airspace. Air Carrier X never entered DKK airspace. The DKK Sector handles all flight plan information; RADAR identifies all aircraft; points out aircraft to other sectors; mostly ZYZ Centralia Sector; that the aircraft will enter; hands off to high side; Dansville DSV; coordinates; talks to and issues higher altitudes in order to keep aircraft from leveling and wasting fuel; time and money and to provide a service to the customer. Again; the aircraft routinely never enter the DKK airspace. While I was conducting OJT; the trainee accepted RADAR contact on Air Carrier X; a B737 departing CYYZ for FLL; requesting FL350. Using the interim procedures; we entered an interim altitude of FL230 in the data block and handed off to Dansville sector (DSV). DSV entered an interim altitude of FL250. After this occurred; the trainee called the Centralia Sector and pointed out Air Carrier X climbing to FL250 Southbound. Centralia accepted the point out. The aircraft then checked in on our frequency and was given a climb to FL250. Air Carrier X was then switched to DSV frequency. Shortly thereafter; DSV controller called us and asked if we had gotten higher from Centralia Sector. Trainee responded; 'Yes; we did.' DSV climbed aircraft in Centralia's airspace and Centralia reported it. Based on the trainee's response and why he gave it; my reaction at the time that it was a correct response; and the question as to why the DSV controller called to ask us this question... I believe the question was answered correctly; but not asked correctly. If the question would have been; 'Did Centralia release higher; or approve higher'; the answer would be no. But the question was asked and interpreted by the trainee and me as if asking 'Did you actually get higher'; because I'm surprised Centralia accepted the point-out because of traffic or Air Carrier X checked on frequency only climbing to FL230 and DSV was checking to make sure it was OK to climb to FL250. Recommendation; DKK sector routinely works aircraft that never enters DKK airspace. Recommendation; change this procedure; the more people involved in coordination; the greater chance of something going wrong.