MidloDad,
Your understanding of the FED written rule is fine and the rules in FED, OBR, and NCAA all read the same. But...
As a number of umpires here have told you, authoritative opinion and interpretation allow the fourth out against B/R when he doesn't reach first base.
Here's the history as I understand it. In early 2001, a huge debate occurred on Internet message boards about this problem. One side followed a literal interpretaion of the rule (the run scores) and the other followed the example play given by Rick Roder in the Jaksa/Roder manual. The J/R manual is based on the teaching notes of one of the two professional umpiring schools, and it is reasonable to expect that a significant fraction of today's MLB umpires would follow the dictates of J/R. One of the key debaters was Carl Childress, who is the author of the BRD (Baseball Rule Differences). [In the second post of this thread, MST made an inside joke by referring to CC as "a guy from Texas".] He (CC) argued strongly for a literal interpretation of the OBR rule. However, he was good enough to ask Mike Fitzpatrick, the director of the PBUC (minor league umpires) what his ruling would be, and the answer was to allow the fourth out. Writing about it later in the BRD, Childress said "Color me ... surprised. I mean astonished."
He followed up with the NCAA, leading to the following interp, which makes fairly loose usage of the term "force out":
NCAA Rule Clarification 4/18/01 8-6-a, b
PLAY: Two outs, runners on second base and third base. The batter singles to the outfield, but injures himself coming out of the box and cannot continue to first base. R3 scores easily. R2 is thrown out at the plate for the third out. The catcher then throws to first base for a fourth out on the batter-runner. RULING: This would be considered a live ball appeal. The out at first base would be considered an advantageous out for the defense and the very fact that they made the play would indicate their choice of this fourth out. Since the batter-runner was out on a force out at first base, R2’s run would not count."
In May, 2001, Elliot Hopkins, Rules Interpreter for NFHS, made the same ruling.
So far as I know, none of the rule sets have explicitly included these interpretations in the rules, or casebooks. Nor have I read any opinion from Jim Evans, the head of the other umpire school, although he surely has an opinion.
I don't like the interpretations, and I don't like having clearly written rules which are at variance from accepted practice. But variance between written rules and actual practice is common in baseball, and we need to accept that which is commonly accepted. Fortunately, fourth out situation are rare.
There are more important rules which need rectification. Anybody want to sign a petition to abolish the DH?