Air carrier departure from RSW was questioned by ATC regarding flight track; reporter alleging runway heading was assigned; ATC claiming CSHEL TWO SID was assigned procedure.

Narrative

Fort Myers Clearance Delivery cleared us to Orlando via the CSHEL TWO Departure; direct KEVYN; and direct MOANS; as filed. The active runway was later changed from Runway 6 to Runway 24. When checking in with Tower; I asked my First Officer to clarify our departure instructions as a Runway 6 departure depicted radar vectors; as we expected from our clearance; but a Runway 24 departure did not. Tower cleared us for takeoff; but said he had just dropped our flight plan; and to stand by. He then cleared another aircraft to land on the same runway we were holding in position for takeoff. He came back and told us to fly a 238 heading and expect vectors to KEVYN on the CSHEL TWO. With this confirmation; we took off and maintained a 238 heading. We checked in with Departure Control; reporting a 238 heading; climbing to 4;000 FT. We were told to climb to 10;000 FT. Shortly after; the Controller asked us if we were on the CSHEL TWO; to which my First Officer responded negative that we were on a 238 heading. The Controller then turned us to a 360 heading and a level off at 4;000 FT. She then gave us a climb to 10;000 FT on a heading of 360; along with a phone number to call for a possible flight deviation. At no time did we receive a TA or RA; nor did we see any traffic on TCAS. The clearance we received from Clearance Delivery made sense for a Runway 6 departure because it instructed radar vectors to join the departure and was consistent with what was depicted on the SID. When the runway was changed to Runway 24; the vectors direct to KEVYN became inconsistent with what was printed on the departure. This is why I asked my First Officer to confirm our departure instructions with Tower. When Tower told us to fly a 238 heading and expect vectors to KEVYN on the CSHEL TWO; I was satisfied we were complying with our assigned instructions. In hindsight; I wish I had taken the time to demand more specific verbiage from the Tower to clarify that we were in fact to fly a HEADING versus the RNAV departure. The fact that we had another aircraft cleared to land on the runway we were delaying takeoff on; in order to clarify departure instructions; made me feel anxious to quickly confirm. I felt the terminology used by Clearance Delivery; especially given the runway change; left doubt as to whether we were to expect vectors or to fly the departure as depicted. But I thought that Tower had cleared up the confusion. I was wrong. It may have helped had we made time to clear the confusion with Clearance Delivery before we got to Tower when the pressure was on to take off promptly because of the aircraft cleared to land behind us. However; a simple; 'Fly the CSHEL TWO;' from Clearance Delivery or Tower (without the direct KEVYN from Clearance Delivery; or the fly heading 238 and expect vectors to KEVYN from Tower); would have been crystal clear. As always; it is the crew's responsibility to resolve confusion. Unfortunately; we thought we had.