Does the run count???

[EHLNOLA]

4.09 HOW A TEAM SCORES.
(a) One run shall be scored each time a runner legally advances to and touches first, second, third and home base before three men are put out to end the inning. EXCEPTION: A run is not scored if the runner advances to home base during a play in which the third out is made (1) by the batter-runner before he touches first base; (2) by any runner being forced out; or (3) by a preceding runner who is declared out because he failed to touch one of the bases.

By the above verbiage it leads me to believe that the runner was not forced out. Which then leads me to believe that the run scores.

A FORCE PLAY is a play in which a runner legally loses his right to occupy a base
by reason of the batter becoming a runner.
Rule 2.00 (Force Play) Comment: Confusion regarding this play is removed by remembering that
frequently the “force” situation is removed during the play. Example: Man on first, one out, ball hit
sharply to first baseman who touches the bag and batter-runner is out. The force is removed at that
moment and runner advancing to second must be tagged. If there had been a runner on third or second,
and either of these runners scored before the tag-out at second, the run counts. Had the first baseman
thrown to second and the ball then had been returned to first, the play at second was a force out, making
two outs, and the return throw to first ahead of the runner would have made three outs. In that case, no
run would score.
Example: Not a force out. One out. Runner on first and third. Batter flies out. Two out. Runner
on third tags up and scores. Runner on first tries to retouch before throw from fielder reaches first
baseman, but does not get back in time and is out. Three outs. If, in umpire’s judgment, the runner from
third touched home before the ball was held at first base, the run counts.

My thoughts are this:

The runner was forced to second base. The 3rd out did not come as a result of a force play. Yes, the runner was forced to second but he was not called out for not beating the ball to the bag. The fielder did not use the force out to end the inning. He tagged the runner. Seems to me that by opting out of playing the force he went to his second action which was to tag the runner.

I may be way off base, but that's ok. I'd rather know for sure than think I know for sure.

I'm actually a basketball official that decided to umpire baseball for the first time this season. So I'm still learning all of the rules and ins and outs if you will.

Thanks
Earl