Declared out vs. put out

[greymule]

Just to make sure, I want to review something we discussed a couple of years ago. This is OBR:

Bases loaded, 2 out. On a 3-2 count with the runners going, ball 4 gets away from F2. The runner from 2B rounds 3B, and F2 throws to put that runner out before the runner from 3B crosses the plate. Ruling (straight from the OBR book): Three outs, but the run from 3B is still permitted to score on the award.

However:

Bases loaded, 2 out. Ball 4 gets away from the catcher. The runners are in no hurry, but the batter, thinking there's a chance for 2 bases, sprints to 1B and rounds the bag, passing the runner on 1B before the runner from 3B crosses the plate. Ruling: Three outs, no run, because the BR was "declared out" rather than "put out."

Is that correct?