If you define it the second way, the doctor hadn't completed boarding. (And I don't think it's a totally ridiculous torturing of the words to read it that way.)
1. Airlines in general don't seem to have defined 'boarded' in their contracts. This usually means that the meaning as tested in court will default to that commonly understood, i.e. 'has entered the aircraft and is seated'.
2. Accepting the manifest isn't final, there have been many instances of aicraft returning to the gate for manifest changes. Even to embark a late but high-value passenger. So when is the aircraft finally boarded? when the wheels leave the ground?
3. The United CEO stated that the flight in question was 'fully boarded'. That may have been an error in terms of hsi legal team but shows that even the airline doesn't have a concrete internal definition.