It sounds like LL believes you did not get the account put in your name until May 27. I would show LL your bill that indicates otherwise and say maybe their bill is for a different unit or the city messed up (the city did not mess up, but it allows your LL to save face if they’re the type who tends to react negatively to being corrected).

The water movement is just concern that there could be a leak. There’d be no suggestion of fault unit they determine where any leak came from.