What is the correct equation for rotor solidity?
Rotor solidity is the area of the rotor disk that is actually occupied by blade area.

Area of rotor blades = $N \cdot c \cdot R$

Area of rotor disk = $\pi \cdot R^2$

$$Rotor \ solidity = \frac{N \cdot c}{\pi \cdot R} \tag{4}$$

So all three of the equations OP found would not be usual. Good reference books are:


Helicopter Performance, Stability and Control by Raymond R. Prouty
Principles Of Helicopter Aerodynamics by J. Gordon Leishman.


Both these books use equation (4) for rotor solidity. It is possible for a factor 2 to pop up in the coefficients which are usually considered regarding disk area and solidity: $C_T, C_P$ and $C_Q$. From Leishman: