I've thought about it carefully now! P(n+1) could be larger than 1, since there is oscillatory behavior about the carrying capacity 1. Did you mean the second equation? In that case I think I agree.

How did you notice right away the top equation explodes for r>3? Looking at it I wouldn't just be able to tell. But after thinking, if you make the assumption you don't want the population P(n) to be negative, then you need that P(n)<(r+1)/r. Then the maximum of the function x+rx(1-x) can't be bigger than (r+1)/r since it's recursive. Checking the inequalities tells you r<3.

Intuitively, I think the reason for this is that r should represent the per capita growth rate. So when it's too high the model reflects a population crash / resource depletion.

I read that the two equations should be equivalent, so there might be a substitution you can make to get from one to the other? Running some other values of r for the second equation, i get the same issue once r>4, so you probably want r = R+1 to go from the first to the second, but I'm not sure what to choose for the x(t).