How many critical points? Why? Critical points are defined (by wikipedia) as the "end-point of the phase equilibrium curve". 

I would say this diagram therefore has three critical points. But this is unnatural. I doubt its true.

Other curves are different from the ones with critical points because "they can run to infinity". But I think thats ridiculous because its physically impossible. 

I don't know if I'm interpreting PDs correctly but I think that what defines that you are on a phase in these diagrams is that if you travel along a horizontal line you meet with a phase equilibrium curve.

If you don't then you were either on a single phase the whole time or on a sort of "critical phase".

I don't like it though. For example if you travel along a horizontal line that wont intersect the L-V curve, how would you know when the LV critical phase is present? Regions are not clearly defined here, except for monoclinic.

Whats wrong with my ideas?

How many critical points does this diagram really have and why?

Thanks.
This phase diagram has one critical point - it is where the line separating the liquid and gas comes to an end - it is now one fluid. So, you can go from liquid to gas (or vice versa) by swinging T,P around the critical point without undergoing a first order phase transition.

The points with arrows pointing to them are triple points, which are also special points, but different from the critical point. Movement away from a triple point means that one (or more) of the phases have to undergo a first order phase transition to the stable phase(s).

Thermodynamically, these are very different things.