It may be that the customer relations agent had as much problem as I had in trying to unravel all of that (you certainly had a very difficult travel journey here). Ultimately what matters is the flight that you were last booked on relevant to the claim, and the one you ended up using. I think that's BA188 on 7 June versus UA161 on a date that isn't clear, and it depends on original, scheduled and actual arrival time. It may be that you have more than one claim here, at least on paper, but perhaps it's best to treat it as one claim.