The correct expressions for the electrostatic force, $\vec{F}_1$, and the magnetic force, $\vec{F}_2$, acting on a charge $q$ moving with velocity $\vec{v}$, are given by the Lorentz force law. This law states that the total force acting on a charged particle in both an electric field and a magnetic field is the sum of an electrostatic force due to the electric field and a magnetic force due to the magnetic field.

The electrostatic force is given by:

$\vec{F}_1 = q \vec{E}$

where $\vec{E}$ is the electric field.

The magnetic force is given by:

$\vec{F}_2 = q(\vec{v} \times \vec{B})$

where $\vec{B}$ is the magnetic field, and $\times$ denotes the cross product, indicating that the magnetic force is perpendicular both to the direction of the velocity of the charge and the direction of the magnetic field.

Therefore, the correct option is:

Option C: $\vec{F}_1=q \vec{E}, \vec{F}_2=q(\vec{V} \times \vec{B})$

Options A, B, and D are incorrect because they do not accurately reflect the definitions of electrostatic and magnetic forces as described by the Lorentz force law.