Two different non-relativistic limits of the Klein-Gordon equation

Question

The first one is the Schrodinger equation. This limit is discussed in books and this Phys.SE post and is obtained by plugging in $$\phi=\psi e^{-imc^2t/\hbar}$$ into the KG equation and then ignoring some terms.

One other limit is discussed by Ron's answer here. This limit just throws away the $\frac{1}{c^2}$ term from the KG equation. Honestly, this limit makes more sense than the previous one because it gives you the familiar Poisson equation from Newtonian mechanics $$\nabla ^2 \phi=0.$$

My question is, how can these both be the non-relativistic limit of the Klein-Gordon equation? Are there ambiguities in taking a non-relativistic limit? What's the general issue here?

Also, the second limit is well-known for Maxwell and General relativity, giving you the Coulomb and Newton theories. Does the first limit make sense as a non-relativistic theory of Gravity and electromagnetism?

Answer 1

I am now thinking that the first limit is the general limit, while the second limit is a special case for massless fields.

The Schrodinger equation looks like :

$$m \frac{ d \phi}{dt}=\frac{1}{2}\nabla ^2 \phi$$

If we set $m=0$ on the LHS, we get Poisson's equation.

So it seems like the Schrodinger wave equation is the general equation of massive Newtonian force fields. I'm not really sure, so feel free to correct.