Do the Bullet Cluster remnants prove that dark matter consists out of particles? Some time ago the remnants of a collision between two collections of stars were discovered. These remnants are called the Bullet Cluster:

The color blue (non-visible in reality) represents the presence of dark matter. Pink represents the normal matter (I'm not sure if the color pink in this picture is actually visible, contrary to the blue).
There are some theories around to explain the nature of dark matter. They can roughly be divided into two groups:
1)Theories that state, dark matter consists of particles
2)Theories that modify gravity (apart from MOND) itself, for example, the theory propagated by the Dutch physicist Erik Verlinde (who's "contraption" here in the Netherlands was announced on TV as a huge breakthrough and for he received the Spinoza Prize. i.e. a lot of money, in 2011; much too much, in my opinion) which see gravity as entropic.
Theories of a modified gravity make the prediction that dark matter is tied to normal matter. Now in pictures of the bullet cluster, it seems like matter and dark matter are clearly separated.
Does this prove that dark matter has to consist out of particles?
 A: Yes. Kind of.
Let me first clarify something. Where you write

1)Theories that state, dark matter consists of particles

I would have written

1)Theories that explain this by additional matter, called dark matter

The reason I'm being picky is because there's dark matter models where it's made out of solar-mass-objects, many think it's made out of lead-atom-mass particles (e.g. WIMPs), and some think it's made out of a more ether-like space-filling wave (e.g. Axions). All those three would equally fall under (1) as they are additional mass, as opposed to your possibility (2) which instead modifies the laws of gravity. Further, within quantum mechanics, "particle" and "wave" is two sides of the same coin anyway. With that picky hat on, I'd therefore rephrase your question as "Does the Bullet Cluster prove that dark matter consists of additional mass [as opposed to modified gravity]?"
Now, the original paper indeed answered that question with a clear "yes". That's because, as you write, one would expect any modified gravitational law still to be centered on the baryonic ("normal") mass, but in the bullet cluster (and dozens of other mergers that have since been studied) that's not the case. That's why I answered your question with "yes".
But then why did I add the "Kind of"? Well, because this is science, and in science, one can never conclusively proof anything. Indeed you can add wiggles to your theory of modified gravity to explain why it wouldn't be busted by merging clusters. For example, you can try additional "unseen" matter, or argue that in an ideal symmetric case you might get such an offset as an artifact. Or, you can use Verlinde's excuse and develop a theory that by construction only applies to equilibrium situations, thus a priori taking merging clusters (and the Cosmic Microwave Background) out as possible evidence against your theory.
Taken together, it is not surprising that almost all scientist work with the dark matter hypothesis, and modified gravity as an explanation for dark matter really is marginalized and worked on by but a handful of scientists. (Though of course many work on modifications with gravity as such, that's well-motivated for many reasons, e.g. we don't have a theory of quantum gravity)
