It is clear that the joker knows that Batman is not Bruce and I really thought that he was gonna have a really bad reaction, like going on a killing spree and crazy the instant that he realized that Bruce was death (or at least out of his reach to kill). It really stroke as a surprise. Maybe he is waiting to go crazy and he is gonna burn down the city.

This is a common misconception about the Joker. He does not want to kill Batman, in fact he can't kill Batman even with unlimited cosmic power he can't do it. The Joker has likely known who Batman was in real life for years and yet he never directly attacked Bruce because Bruce is the Mask that Batman wears to the rest of the world. The Joker likely knows that Dick is now Batman but does not care because Batman just switched his Mask from Bruce to Dick. The Joker doesn't see the person, he sees the bat.  Whoever happens to be the Bat, so long as they fill the shoes nicely, he will vex.

I know that he doesn't really wanna kill him but he likes fighting the other and definitely he is the one who wants to defeat him, even if that doesn't mean killing him.

It's very simple. For the Joker it's Bruce Wayne or GTFO. 

No, the Joker doesn't see Bruce he sees the Bat. Bruce for the Joker was simply the Mask that Batman wore around people, as long as Dick can be as incorruptible as the Bat then the Joker will simply think the Bat has switched masks.

I re-read it earlier and he did say "No matter how many I killed, Batman, my Batman, was gone."

" This is a common misconception about the Joker. He does not want to kill Batman, in fact he can't kill Batman even with unlimited cosmic power he can't do it. The Joker has likely known who Batman was in real life for years and yet he never directly attacked Bruce because Bruce is the Mask that Batman wears to the rest of the world. The Joker likely knows that Dick is now Batman but does not care because Batman just switched his Mask from Bruce to Dick. The Joker doesn't see the person, he sees the bat.  Whoever happens to be the Bat, so long as they fill the shoes nicely, he will vex. "

I think it does matter which Batman it is to a point. Bruce will always be the best Batman ever. For the Joker, Dick might pose a challenge but never one a great as the original Batman.

But then agian in the next issue, he might change his personality again to fit the situation and be totally fine with Dick being the new Batman. You never know with him....

For Joker he probably thinks, "Why battle a second rate batman, it just doesn't compare to battling the best of the best." 
 
It's like eating the best food in world, food made only for gods, why eat anything else....it just doesn't compare.