Advance ratio formula for drone propeller in vertical climb
I could not find equation (2) in my copy of J. Gordon Leishman' Principles of Helicopter Aerodynamics. Some relevant info on momentum analysis in forward flight can be found here on page 63.

Advance ratio as a useful measure of blade twist is less meaningful in helicopters, due to the much greater variety in the direction of the velocity vector. It makes more sense to optimise blade twist for forward flight than it is to optimise for vertical flight, a state with much higher fuel consumption which helicopters do not particularly like.

If you would like to use the advance ratio for helicopters as stated in the wiki article, it makes more sense to use the helicopter one. There may be a direct parallel between the states you describe in your question, but for the helicopter case it is not the design state while for the fixed wing it is.

Whether either is relevant to a propellor drone depends on the design of the drone, whether the propellor is shrouded etc. But for a drone, free velocity would usually not be close to perpendicular to the propellor plane, just like in a helicopter.