In simple terms
Speed rating indicates how often a player will "attempt" to steal.
Steal rating indicates how likely a player will be "safe" when they attempt to steal.
Base running rating is basically how "smart" they are on the base path.

IE. The "slowest" runner on the team could have the highest "speed" rating if they attempted the most steals.
The "fastest" runner on the team could have the worse steal rating of they get thrown out a lot. Maybe they get bad leadoffs, etc..I

Also just like real life the runner may or may not attempt to steal it you give them the green light.

Oh.

I thought it was the other way around. Players with a good steal rating are more likely to get a good jump and not have to turn back, and then they are more likely to be safe if they are fast enough to make it. It makes more sense that way. Why would it be the other way? I’m not saying you’re wrong, but I’m wondering about the logic behind it being that way.

As I've understood it, steal rating is more aligned with success rates, speed rating dictates more how often they attempt to steal (how often they get a good jump, etc.) So yes, what you are saying about them more likely to be safe is correct but so is what Reed said about the frequency of them attempting to steal being keyed off of speed. (In other words, think of it like this: some guys attempt to steal very infrequently but pick their spots well and are almost always successful. These type players have low speed ratings- meaning a steal attempt is triggered very seldom- but high stolen base ratings- meaning that when they do attempt to steal they are very likely to be successful.) If you have ever played Strat-O-Matic with their advanced steal system I could explain it perfectly that way too, but I'm not going to assume that is true. :)

EDIT: I probably should have added to, that in my experience part of what makes OOTP so good and deep and nuanced is that it is rare for any one rating to behave completely independently. Ratings generally work in concert with other ratings to produce results. As in this example, where the speed ratings and the stolen base ratings both play a role in the stolen base process, in addition to other aspects like pitcher's hold rating, catcher's arm, etc.

From the manual.

Running Speed
Running Speed is a measure of how quickly a player can run from one base to the next. Players with a high rating in Running Speed are more of a threat to steal bases, and are more likely to advance further on a ball in play. Running Speed is strictly for offensive play. For example, Running Speed does not factor into an outfielder's range. However, Running Speed and Defensive Range are linked internally. For example, as a player ages, his speed and range will generally decline at the same pace.

Stealing Bases
Stealing Bases is a measure of how good a player is at stealing. Stealing bases is not a matter only of speed, but also timing, knowledge of pitchers, and ability to 'get a good jump.' It is possible for a player to have a high Running Speed rating, but a low Stealing Bases rating, and vice versa, although the most dangerous baserunners have high ratings in both. A combination of strategy settings for stealing bases and Running Speed is used to determine how often a runner is given a "green light" to steal a base. Speed and pitcher Hold Rating then determine how good a jump the runner gets. If he gets a good jump and goes (ie a stolen base attempt) the Catcher Arm, pitcher Hold Rating, the type of pitch and Stealing Bases rating are the factors which determine whether the runner is safe.

Baserunning Instincts
Baserunning Instincts is a measure of a player's instincts when running the bases. Players with a high rating in Baserunning Instincts are more likely to take advantage of fielder miscues to advance a base, and less likely to get thrown out due to baserunning mistakes.

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