The definition or point you have made is wrong.
The viscous damping coefficient equals decay constant divided by two times mass
$$\gamma = \frac{c_{2}}{2m}$$
The decay constant or damping coefficient $(\gamma)$ equals the ratio of the viscous damping constant to two times the mass. Note that some authors may use terms differently.
$$\color{red}{\boxed{\text{Decay constant }=\frac{\text{Damping constant }}{2\times\text{mass }};\ \gamma=\frac{c_2}{2m}}}$$
If the magnitude of the velocity is small, meaning the mass oscillates slowly, the damping force is proportional to the velocity and acts against the direction of motion.
$$\large \color{red}{\boxed{F_d=-c_1V=-c_1\dot x}}$$
The constant $c_1$ is called the "damping constant". Both $c_1$ and $c_2$ are essentially the same.
To understand why $c_2=c_1$, consider a damped free vibration. The equation of motion is derived using Newton's second law of motion: $m\ddot x=F_d+F_s=-c\dot x-kx$.
$$\large m\ddot x+c\dot x+kx=0\implies\ddot x+2\zeta\omega_n\dot x+\omega_n^2x=0$$
where
- damping ratio, $\zeta=\frac{c}{c_c}=\frac{\text{damping coefficient}}{\text{critical damping coefficient}}=\frac{c}{2\sqrt{km}}=\frac{c}{2m\omega_n}$
- natural frequency, $\omega_n=\sqrt{\frac km}$
The underdamped solution can be expressed in the form:
$$\large x(t)=A_0e^{-\zeta\omega_nt}\cos\left(\omega t-\varphi\right)=A_0e^{-\gamma t}\cos\left(\omega t-\varphi\right)$$
As you can see, the decay constant $\gamma=\zeta\omega_n=\frac {c}{2m}$. It's also called the damping coefficient by some authors. And $c$ in $F_d=-cv=-c\dot x$ is called the damping constant. One may call $c$ as the damping coefficient as well. So, it's essential to know what notation is used for a term. It depends on the context and authors.
