A330 Captain reported the number two engine was started by mistake. Subsequently; ground crews alerted the pilots to stop engine due to FOD ingestion.

Narrative

This was leg 2 of 2 for the day and we were in the pilot lounge reviewing our flight documents. We went over several items to include MEL's; Operations; departure; weather; and so forth. This aircraft had 3 MEL's related to the APU. Additionally; our destination weather did not look good at the time including wind gusting up to 65 knots and hail from 1' - 1 1/2'. Needless to say there was lots of discussion. We went out to the airplane; meeting the Lead around the steps of the aircraft. I had a brief discussion with him about the need for an air start due to the APU being on the MEL list. I asked him; as we were standing towards the nose of the aircraft looking back towards the tail what engine he would like us to start first. He said the left engine. I clarified with him; because we were looking at the engines from the nose back. I asked him; you mean engine number 2? He agreed and said yes.I paused because its preferable to start Engine number 1 first. I then told him if that makes it easier for you; we can do that and we will plan on starting engine number 2 first. We then headed up the stairs. We setup for the flight; the FO did the exterior inspection; we talked to dispatch for clarification on weather and a plan of action further down the line closer to CVG. We also spoke with [maintenance] for clarification on the MEL's. The FO and I came together to share a mental model of what we were facing. I told our jump seater who is also a FO on our aircraft that I would like if she would speak up if she had any relevant ideas or thoughts as our flight continued. We finished our check and were ready to go. Communication with ground crew was established and confirming that engine number two would be started first. I started engine number 2 and told ground crew the number 2 was spinning. Shortly thereafter ground crew said you are starting the wrong engine and instructed me to kill the start. I immediately aborted the start on engine number 2.To me; this indicated a complete failure in communication between us in the cockpit and the ground crew. It was a request by them that we start engine number 2 first. However; this is not what they wanted and led me to believe that either we were failing to properly communicate or that the ground crew didn't know the difference between number 1 and number 2 engine. I resorted back to basics. I asked the ground what direction he was facing. He said the same way as you. I asked him if he was looking at the airplane and he said no. He said he was looking at the facility in front of the aircraft. I said OK; so if we are looking the same direction what engine do you want me to start first. He said the left engine. I said OK; the left engine from this orientation which was different from what we agreed to on the ground. I told him that was engine number 1; and that I could start that engine first. We the proceeded to start engine number 1 and then disconnected the GPU and the Air cart. During push back the ground crew called and said you are cleared to start your left engine. I said thank you; but we will need to wait until the push is complete and the brake is set. We are going to need to perform a cross bleed start on engine number 2 and I will need to increase thrust in order to perform the start. In order for us to continue with our procedures we need the push completed and the brake set. He acknowledged.Push was complete and brakes were set. We were cleared to start engine number 2. Further on in the engine 2 start sequence ground crew called in a panic and I could not tell what he said. I asked him to repeat what he said. He then said to stop engine start and that the engine ingested FOD. I immediately aborted engine number 2 start and asked him what happened. He reiterated that the left engine ingested FOD. I told him we would need to return to the gate. There was a discussion about towing us back in however; I informed him that we couldn't shutdown engine number 1 because we didn'thave an APU. He asked if we could taxi in on our own power and I said yes. We ran through a checklist to try to organize our procedures before we taxied in. I requested that he let me know when everyone was clear so that I could taxi back to the gate. Once at the gate I called again requesting the GPU due to the inoperative APU. We received external power and then proceeded to shut down engine 1 and run the parking checklist.The doors were open and I went to look at engine number 2 to see what FOD was ingested. I was told while walking to the engine that someone put a laptop on the inside of the engine number 2 nacelle. FOD was everywhere and they were cleaning it quickly. I returned to the cockpit to explain the situation to the FO and make and entry in the logbook.