In most cases mentioned only hot and cold dark matter. But sometimes there is information about warm matter.

To say it roughly, this depends on the mass/velocity of the dark matter particle. In Standard Cosmology, we need cold dark matter, i.e. non relativistic dark matter, but this does not mean we 'only' need cold dark matter. We can have some admixture here.

At the origins, dark matter was introduced to explain the motion of the galaxies. But today, we have many more indirect evidence, as I said in the post. And dark matter needs to be cold, according to the evidence. As I said, non minimal cosmology is possible too, and in this case, you may have hot dark matter as well.

Warm dark matter is some form of dark matter whose speed is intermediate.

Do you agree that there is one more, intermediate state?

No. I neither agree nor disagree. You have models with a single dark matter species, and models with several of them. Both are allowed so far.