Exact meaning of constant volume heat capacity From Wikipedia:

$$
\left(\frac{\partial U}{\partial T}\right)_V = \left(\frac{\partial Q}{\partial T}\right)_V = C_V,
$$

$C_V$ is what known to be constant volume heat capacity. I don't really get the exact meaning of 'constant volume'. Does it mean that (a) the whole process takes place under a constant volume, i.e, it can be any volume as long as the volume does not change during the process, the volume is maintained at its initial value throughout the process, or (b) the $C$ is measured under a fixed constant volume $V$?

If (a) is correct, are the rates of change of $Q$ with respect to $T$ same at, for example, $V = \pu{1 dm3}$ and $V = \pu{2 dm3}$? We are not considering only perfect gases.

If (b) is correct, what is the value of $V$ for standard $C_V$?
(a) is correct. (b) is wrong. $U$ depends on $V$ only in the combination $v=V/N$. $v$ is the specific volume, an intensive quantity that does not depend on the size of the system. 

We can derive this from first principles. Take $U = U(T,V,N)$. Because $U$ is extensive, $$U(T,\lambda V,\lambda N) = \lambda U(T,V,N) \quad \Longleftrightarrow \quad U(T,V,N) = \lambda^{-1}U(T,\lambda V,\lambda N),$$ by taking $\lambda = N^{-1}$, we can write this expression as $$U(T,V,N) = NU(T,V/N,1) =: Nu(T,v),$$ which produces the desired functional relationship.