Runge-Kutta and Butcher table?

In the Wikipedia article on Runge-Kutta methods, there is a notation explained using a Butcher table with a $c_{i}$ vector (nodes), a $b_{i}$ vector (weights) and a runge-kutta matrix $a_{ij}$.

My question is : does every runge-kutta-something method is entirely summed up by this Butcher table, or is there some subtleties ?

For example, if we take the Feagin RK12 and RK14 methods explained here :

and with the coefficients :

do the coefficients completely constrain the numerical scheme or is there additional details to put in the integrator ?

For example when we say RK14(12) which is a "14th order method with an embedded 12th order method", can I simply put the Butcher table in a generic RK integrator which takes $c_{i}$, $b_{i}$ and $a_{ij}$ as arguments, or is there some additional details to know ?