Corporate captain reported confusing ATC clearance for takeoff resulting in a cancelled takeoff clearance and subsequent taxi clearance back for takeoff.

Narrative

We were departing from John Wayne/Orange County Airport (SNA) on an Instrument Flight Plan; where we experienced a takeoff cancelation clearance from tower on Runway 20R and Tower instructed Aircraft Y to go around. After engine start; we contacted ground for taxi instructions and were advised to taxi to Runway 20R via Bravo and Lima. In route to Runway 20R; ground instructed us to monitor tower frequency. The Pilot Monitoring switched over to the tower frequency. As we were holding short of Runway 20R; tower advised us to be ready to go for a quick departure between a landing aircraft and a approaching to land aircraft. Immediately after the first plane landed; tower instructed us to line up and wait and be ready for an immediate takeoff due to the approaching aircraft. As we were waiting for a takeoff clearance; tower was instructing the aircraft that landed to exit the runway without delay; as we could tell the sequencing was getting very tight. After I lined up the aircraft for departure on Runway 20R; I could see the landed aircraft was still on the runway. Tower again advised us to come up on power; which was something we have never heard tower use as standard phraseology. Due to this non standard phraseology; both I and copilot thought tower advised us to up on tower (meaning the tower frequency) even though we thought we were already on tower frequency. We question each other and asked was that power or tower? Over the years; both the pilot flying and pilot monitoring have heard tower use the phraseology to come up on tower frequency but; never heard a controller use the term 'come up on power'. We have previously heard controllers use; 'be ready for an immediate takeoff clearance or be ready for an immediate departure' as standard instructions from Tower. Due to the confusion and the rushed nature of trying to get us out before the approaching aircraft; we thought we were still on ground frequency by mistake and became confused about the instructions. My pilot monitoring instinctively reached over to the radio and switched the frequencies from the active to the standby; which was no longer ground but the departure frequency. Our standard operating procedure is to place the departure frequency in the secondary on COM 1 under the tower frequency; so in the busy climb portion of the departure all we have to do is to push one button to switch between the tower and departure frequency. However; since the pilot monitoring hit the button we were off tower frequency for approximately 1 to 2 seconds. During this short period; we must have been given the take off clearance; which we never heard. I instructed the pilot monitoring switched back to the tower frequency; as that was the departure frequency in the active. We were instructed to cancel the take off clearance; exit the runway immediately via left turn taxiway K; then stop on Runway 20L; and Aircraft Y was instructed to go around. I complied with tower instructions immediately and exited the runway. The pilot monitoring advised tower that we were uncertain as to the instruction to come up on power and thought the controller instructed us to come up on tower (meaning tower frequency); which was the cause of the confusion. The controller advised it was okay and that they would sequence us back in for takeoff. After a few minutes; Tower instructed us to make a left turn on Runway 20L; left on Lima and hold short of Runway 20R. I was then cleared for takeoff with no further incident.