7.12: Unless two are out, the status of a following runner is not affected by a preceding runner’s failure to touch or retouch a base.
The runner on first (the following runner) stole second while the preceding runner tried to steal third. So, why would anyone other than Segura be out caught stealing in a rundown? He obviously could not be allowed to be safe at first (if that had happened) because then the runner on first would have batted before the runner who was on second.
Way back when, Germany Schaefer was on first with a runner on third and a delayed double steal was called. Schaefer stole second but the catcher held the ball instead of throwing it so Schaefer then stole to set up the double steal again. That time it was successful. I thought that caused a rule change to prevent reverse steals.
7.03 is the actual rule in play here:
(a) Two runners may not occupy a base, but if, while the ball is alive, two runners are
touching a base, the following runner shall be out when tagged and the preceding
runner is entitled to the base, unless Rule 7.03(b) applies.
(b) If a runner is forced to advance by reason of the batter becoming a runner and two
runners are touching a base to which the following runner is forced, the following
runner is entitled to the base and the preceding runner shall be out when tagged or
when a fielder possesses the ball and touches the base to which such preceding run-
ner is forced
The rule you state would be in play if, say, Segura had rounded third and then run abck to second without touching while Braun was on first. Then Segura would be out and Braun would remain at first.
Instead, since Braun was not forced to go to second (remember he went there while Segura was in the run down), the base still technically belongs to Segura, therefore Braun is the one called out.