Can you spot the Lucena position that does not win?

Lucena position gives the superior side a winning endgame in Rook endgames except when it comes to a Rook pawn. When you have a Rook pawn, Lucena position is not going to guarantee you a win.

I have shown two positions here. It is white to play in both positions.

Which is the position that is winning for white? Please do explain the winning idea

The detailed solution is available here.

The first position is wining because Rc3 and Rc7 followed by Rb7 gives you wining. While in second position the black king do not allow you such a manoeuvre.

Yes.Cool.

That's right!

If you play Rc3, Ke7 and Rc7+, after Black has played Kd6 it's a draw. White has to play Rc3 and then Rc8 in order to win. Black rook has to move away from the b file, then white king goes to b7, and when black has run out of checks, white queens and wins.

I meant Rb8.

Or to put it more simply: Rc3, Rc8 , Rb8 is the right way to do ut.