If a soldier gets injured or ill during an APFT, is that considered a fail?

Yes it is a fail. But depending on the circumstance and the Soldier in question, the commander has some latitude on how he can address it. Example, during 2 mile run, someone drops of a cardiac arrest. Fellow soldier stops and assists with CPR and fails event. Technically that is a fail....but I seriously doubt the Commander is going to initiate a flag. Would likely let them retake it at soonest opportunity, particularly if the individual in question has no issues with APFT.

The soldier is a test failure, it is sometimes difficult to apply the standard, especially if the soldier who normally passes the test. As leaders our obligation is to apply the standard and set up the training and coaching needed for the soldier to be successful next time.

You already have your answer and you answered it yourself. The FM states that he failed to meet the minimum passing score so he is a failure. If you are questioning whether or not he should be flagged I understand and I am sure your command will supply that answer shortly. However, per regulations he failed the APFT.