Captain reported being informed of a tail pipe fire and originally thought it was the #1 Engine; instead of the #2 Engine. Airport CFR responded and the Captain dry motored the engine to extinguish the fire on the #2 engine.

Narrative

We landed on [runway] XYL in ZZZ; exited right on [runway] XX; taxied back Taxiway 1 to [the gate area] where I shut down the right #2 engine; continued taxi into [the gate]. Set parking brake; shutdown left #1 engine. Went through hand signals with Marshaler with nothing of note and ran the Parking Checklist. Once the Passenger entrance door was opened; a Ramp Agent came into the cockpit and showed the F/O (First Officer) and I a video of what appeared to be a tailpipe type fire. The F/O and I hurried out to see. For whatever reason; I believed the tailpipe fire to be on the #1 engine as we walked out the jetbridge door. I noticed there was a lav truck already hooked up to the aircraft. I asked the F/O to have the truck disconnect immediately and for the F/O to clear the area. Not certain he heard the last part. I was satisfied we didn't have to alter a normal deplaning of the Passengers; but I expressed to the A-F/A (Flight Attendant) that we should expedite the process of normal deplaning. Then I went to the QRH and ran it. Believing the area around #1 engine to be clear of people; I began the process of dry motoring per the QRH. I heard Folks hollering to get my attention; so I terminated the dry motoring. Apparently the engine in question was the right #2 engine! When I looked out the front right galley door I saw the F/O and Airport Personnel standing on the ramp and I noticed the belt loader hooked up to the rear right of the aircraft. Not sure why the Ramp Personnel would allow this if there was concern the right #2 engine was on fire. Anyway; I asked that the belt loader and personnel on the belt loader be moved immediately and the area cleared. Once satisfied the area was clear; I dry motored the right #2 engine for one minute in an effort to put out any residual fire outside the parameters of the engine. At no time were flames shooting out of the tailpipe. At no time did I see engine instruments out of what I would expect on a normal shutdown and I confirmed both start levers were fully seated in cutoff prior to any motoring per the QRH. The Fire Department ran their tests on the engines and found the temperatures of both engines to be satisfactory with no further cause for concern. Initially I believe I heard the Ramp Agent tell me it was the left engine; which I believe was on his left as he looked forward. Apparently the F/O did not hear that yet did not question me when I asked him to have the lav truck disconnect and move to clear the #1 engine. Obviously communication broke down here. Perhaps if the Ramp Agent said 'F/O side'; or used a headset connect; or knew the tailpipe fire hand signals things would've been clearer or quicker to transpire. Perhaps I should've asked or double checked to confirm the engine he was referring to. I felt time was of the essence initially but in retrospect may have done better by going on the ramp myself to assess the situation. I certainly would've determined the engine with the problem and may have decided that dry motoring was not necessary. But with people still aboard and a potential fire; I opted to run the QRH for what we thought we had and stay in the cockpit/galley to direct any further action that may be necessary. Apparently someone on the ramp called to roll the fire trucks as well without my being informed. I'm ok with that; to a degree; because he communicated as much and that allowed me to focus on assessing the safe deplaning of our Passengers. The Passengers deplaned as normal. I called Dispatch; spoke to the (Operations Chief Pilot) to advise of the occurrence seeking counsel for any further actions I must take. I spoke to Maintenance Control and Line Maintenance and made a logbook writeup. Line Maintenance did their check on the engine and no damage was noted. The aircraft was returned to service and we flew into ZZZ with no further issue.I need to be certain that when another Employee reports time sensitive information that I take the time tobe certain they clarify exactly what they're saying. Not sure; even now; how I believed the engine in question was the #1 and the F/O understood it to be #2. Clarity was somehow lost initially. I believe if the Ramp Agent says possible tailpipe fire F/O side of aircraft then perhaps I send the F/O out immediately to investigate. No doubt if the Ramp feels it's important enough to call the Fire Department before informing me; then it should be important enough to keep that engine in question clear of all Ground Personnel and not let a belt loader and personnel start off loading baggage