Sac fly rule

Question
Question is about whether to credit a sac fly.
Here was the scenario: Runners on second and third, one out. Batter at the plate hit a towering, deep fly to right center almost to the wall. The runner on third was an inexperienced, probably very nervous high school sophomore, put in to run the bases in a varsity game. He had taken a lead, and when the ball was caught, he was for some reason several steps down the third base line towards home. The outfielder threw to the relay man near second base as R3 ran back to third to tag up. After tagging up, he turned around and started for home, hesitated and seemed to wonder whether to run home or go back to third, but then saw that R2 was heading for third, so he started sprinting for home again. The relay man had thrown to third, hoping for a play there on R2 or R3, but no play there. The third baseman then fired the ball to home. The throw was off line a bit and the catcher had to pull off the plate to the first base side to make the catch. The ball hit his glove and he dropped it. The runner scored. Unclear whether there would have been a play at the plate if the throw from third had been perfect.

The controversy: Should this be scored as a sacrifice fly? I thought without doubt it should be a sac fly, but another parent had a different opinion.  Here's his analysis:

If the 3B runner had simply tagged up after
the fly ball was caught and then scored it clearly would have been a sacrifice fly.  However, this play did not occur in a vacuum and that is not what happened. The fly ball was caught while the third base runner was down the line. He returned to third and tagged and then started to proceed home.  He then was either called back by the third base coach or decided it was too late, on his own, and attempted to return to third. Either way, it was his clear intention to attempt to return to third and not
attempt to score as a result of the fly ball into the outfield.  As the third base runner was attempting to return to third he saw that the runner behind him had advanced from
second and was about to be at third.  Third base runner then had no option but to attempt to go home.  If he didn't one of the two runners would be out.  By this time the relay man had the ball (still in the edge of the outfield) and attempted to throw
the runner out at home.  The throw was a bit off line and the catcher dropped it. I would rule that the runner scored on either an error on the throw or on the catcher.  
My main argument for why not to credit a sac fly is because the runner's intent was to attempt to return to third and not
score on the caught fly ball.  Once he attempted to return to third after initially tagging and starting home, any score after that point did not result from the fly ball to the outfield.  He scored as the result of poorly played defense after a base running blunder.

SO, sac fly or not????

Answer
Pat,

Here's the official clause from the rulebook on a sacrifice fly:

"(d) Score a sacrifice fly when, before two are out, the batter hits a ball in flight handled by an outfielder or an infielder running in the outfield in fair or foul territory that
(1) is caught, and a runner scores after the catch, or
(2) is dropped, and a runner scores, if in the scorer抯 judgment the runner could have scored after the catch had the fly been caught."

Given that, I'd score the play a sacrifice fly unless it was clear that 'ordinary effort' would have retired the runner.  In that case, no sacrifice fly and an error on the thrower.

Hope this helps!

Brian