K°P vs KP in equilibrium thermodynamics
The equilibrium constant should be dimensionless, it has to be to be used in equations such as the free energy for reaction at equilibrium 
$\Delta G^0_R = -RT\ln(K_p)$.

If the reaction is 
$\ce{\alpha A + \beta B \&lt;=&gt;\ \gamma C + \delta D} $
 if we write

$$K_p= \frac{P_A^{\alpha} P_B^{\beta}}{P_C^{\gamma}P_D^{\delta}} $$

where $P_a$ etc are partial pressures at equilibrium then $K_p$ is going to  have dimensions in general depending ion the values of $\alpha , \beta $ etc. as you state in your question.

In the past it was sort of implicitly assumed that each pressure was divided by 1 atm so then $K_p$ becomes dimensionless (or the units were simply ignored and numerical values used) but nowadays this is explicitly added so we obtain

$$K_p= \frac{\left(\frac{P_A}{P^0}\right)^{\alpha}  \left(\frac{P_B}{P^0}\right)^{\beta}}{\left(\frac{P_C}{P^0}\right)^{\gamma}\left(\frac{P_D}{P^0}\right)^{\delta}} $$

where $P^0 = 1 \pu{ atm}$  and so makes things dimensionless no matter what the values of $\alpha, \beta $ etc are but does not change any numerical value.

Although I have used partial pressures the same applies to concentrations $\ce{[A], [B]}$ etc where now each concentration is divided by  $\ce{[C^0]}$ where this is  $ \pu{1 mol/dm^3}$.

I'm not quite sure what your notation $K^0P$ etc. means but I suspect it refers to the two cases I have described. Hope this helps.