Consider two parallel plate capacitors, one with a separation between plates of $d_1$ and other with $d_2$, where $d_1,d_2\ll\sqrt A$, and where $A$ is the area between the plates and both in the same medium.
Now let dielectric strength of material be $E$, therefore the maximum possible electric field that can be in the region containing the dielectric is $E$.
Both the capacitors can produce this electric field when charge $ Q=EA\epsilon_0$ is given; therefore the maximum charge that both plates can hold, i.e. capacity of the plates is same.
If $d_1>d_2$, then the potential difference between the plates will be different, which will give higher capacitance of the plates with smaller separation.
Therefore, is the definition that capacitance is the ability to hold charge wrong? Since both plates can hold the same charge but have different capacitance? Should the correct definition be: capacitance is the ability to store charge per unit potential difference?