The fraction $c/d$ is a best approximation of the second kind for a number $x$ if for every other fraction $p/q$ with $q<d$, $|dx-c|<|qx-p|$
Given a number $x$, its best rational approximations of the first kind are those fractions $c/d$ that satisfy $|x-c/d|<|x-p/q|$ for any other fraction $p/q$ where $q<d$.
We are trying to approximate a number $x$ by combining $1/q$ which we do $p$ number of times. If we make $q$ larger, ie., $1/q$ becomes smaller, that means the building blocks become finer which makes the approximation more accurate. But, one can not make the building blocks $1/q$ infinitely smaller, there can be a practical maximum value which $q$ can be chosen. So we need to find the best approximation of $x$ for the chosen value of $q$.
So the idea of the best rational approximations of the first kind is that, $x$ is closer to $c/d$, ie., minimum error, than to any fraction with a smaller denominator.
In the condition for the best rational approximations of the second kind, the error is multiplied by the denominator, ie., $\color{red}{|dx-c|<|qx-p|}$. What is the logic behind setting such a condition to state it is the best approximation of a number ?