Definition of "safe"?
 

There are rules that say, "The batter is out when...", or "The runner is out when...", and it lists all the ways the batter or runner would be out.

I don't see a rule that says, "The runner is safe when..."

What exactly does it mean whe an umpire signals "safe"? Does it mean that the runner has reached the base to which he is trying to reach (or return to)? Or does it mean that the attempted play on the runner was unsuccessful? I've seen umpires signal "safe" while a runner was still between bases.

In one case in particular (NCAA), a runner was trying to score after a fly ball to the outfield. The outfielder throws to the catcher who was on the base path. The runner collides with him as the ball arrives. The umpire signals "safe" even though the runner was nowhere near the plate. Does this mean he had scored? Or does it mean he just wasn't out yet?

Rule 2.00: "SAFE" is a declaration by the umpire that a runner is entitled to the base for which he was trying.

And to actually answer your question... the umpire is saying that on whatever "play" was being made, the runner is safe. He's not necessarily saying the runner made it to a base or not. On any tag attempt (except at home when they miss the bag in certain rule-codes, for no reason that's ever made sense to me), we are to signal out or safe. Also note that we don't always signal safe when a runner does reach a base ... only if there was a play there.

The mechanic that is used to signal "safe" is also used elsewhere on the diamond. "No catch" "No tag" and "That's nothing" are some of the other instances where this mechanic is used.

There are plenty of definitions in the rule book that dictate when a player is "out" if a player is not "out" then he's probably "safe"