Admittedly, I am biased to Eckstein, I had lunch with his agent this past week. To me, Eckstein is one of the very rare few who are beyond student of the game, he IS the living, breathing epitome of the game. After all, the guy literally willed himself to college and pro ball.
Ok, I generally think students of the game are the kids/players who a) watch inordinate amounts of baseball b) watch not just to spectate, but watch to know why things happen and when, what pitches are thrown on which counts and have a grasp of when to bunt, hit/run, as well as, teh fundamentals of technical plays etc. etc.
Also, good students of the game aren't neccessarily great players. I'd like to think I'm one that fits that category. You can study and understand the game and npt be very good. When you combine a great student with a good/great player your getting in pretty rare territory, although most catchers and some pitchers have some of these tendencies. These are the guys who can make the plays instincutally and also know the exact fundamentals of the play and why everyone did what they did.
I think there are plenty of guys who play well who arent students of the game, they just have the god-given repetition re-enforced instincts.
I think there are plenty of students of the game who wouldn't neccarisarily be good teachers or coaches...that takes good communication skills. They exist, but I dont think the two go hand-in-hand automatically.
I do think if your going to teach and coach WELL, you have to be a student of the game. But just cause your a student doesn't automatically mean you will teach and coach well. Make sense?