How many kinds of "weak charges" exist?

The problem with "weak charges" is that electroweak symmetry is spontaneously broken. Before the symmetry breaking, electroweak symmetry is described by an $SU(2)_L \times U(1)_Y$ gauge group.This amounts to three charges: weak hypercharge $Y$ for $U(1)_Y$ and weak isospin (total isospin $T$ and third component $T_3$) for the $SU(2)_L$. Some examples of charge assignments:

$e_L$ (left-handed electron): $T=1/2$, $T_3=-1/2$, $Y=-1$

$\nu_L$ (left-handed neutrino): $T=1/2$, $T_3 = 1/2$, $Y=-1$

$e_R$ (right-handed electron): $T=0$, $T_3=0$, $Y=-2$

$W^1, W^2, W^3$ ($SU(2)_L$ gauge bosons): $T=1$, $T_3=-1,0,1$, $Y=0$

The Higgs mechanism breaks the $SU(2)_L \times U(1)_Y$ symmetry, and only a subgroup $U(1)_{em}$ remains unbroken, which corresponds to electromagnetism. But please notice that $U(1)_{em} \neq U(1)_Y$ (in fact, the electric charge is $Q = T_3 + \frac{Y}{2}$), and that weak interactions alone don't fit in a $SU(2)$ group.So we can't really talk about "weak charges".

You're mixing a few things up here. When you say "three" for the strong force, you're counting the number of colors of quarks, but when you guess "three" for the weak force, you're counting the number of force carriers.

These are two different things. For example, if you counted the number of gluons (the force carriers for the strong force), you'd get eight, not three.

In general, for a gauge group $G$, the number of distinct particles of some type is equal to the dimension of its representation under $G$. The force carrier is always in the adjoint representation, which has dimension equal to $G$ itself, and all matter particles we know of transform in the fundamental representation. So for the three forces, we have:

• Electromagnetism, $G = U(1)$. One matter particle (electron), one force carrier (photon).
• Weak force, $G = SU(2)$. Two matter particles, three force carriers ($W^+$, $W^-$, $Z$).
• Strong force, $G = SU(3)$. Three matter particles (quarks), eight force carriers (gluons).

Then the answer to your question is two or three, depending on what you meant.

I would say two, which is pleasantly consistent with the $SU(2)$ structure of the weak force. One is the coupling strength with the $Z$ boson, and one is the weak isospin which is raised and lowered by the $W^\pm$.