Runner Scores before Batter-Runner Retired?

 
With 2 outs and a runner on third base (R3), the batter swings hard but hits a fair dribbler grounder a few feet in fair territory and a few feet up the third base line.
 
The catcher believes that he has an easy tag out of R3 running home, so he picks up the fair ball and attempts to tag the sliding R3. R3 legally slides under the tag and is safe at home. However, the catcher sees that he still has a chance to retire the slow-runner batter, so he throws to first to retire the batter to end the inning.
 
Does R3's run count?
 
Call
A run does not score if a runner advances to home during a play in which the third out is made by the batter-runner before the batter-runner reaches first base (NF 9-1-1a).
 
In the situation below, although R3 made it safely to home well before the third out of the inning, the third out was due to the batter-runner being called out before he reached first base. In this situation, the run does not count.