So in my book there is a chapter summary that says “nuclear binding energy is the amount of energy that is released when nucleons (protons and neutrons) bind together” and the first time it was mentioned in the chapter it simply called this binding energy. However, when I looked it up online, binding energy is the amount of energy required to separate particles from a system. Is the amount of energy required to separate particles equal to the amount released when they are bound? Or is there something else I’m missing?
Nuclear binding energy can be seen as a "special form" of binding energy. You can see that the given definition for the latter
the amount of energy required to separate particles from a system
also applies to a nucleus: In this case, the particles are the nucleons and the system is the nucleus.
The only difference, as you noted is that in one of the given cases, the binding energy is said to be released, in the other case it is the energy required to break a bond. However, this is no contradiction: If you get $X$ "amounts" of energy when forming a bond between two (or more) particles, you have to "pay" the same amount $X$ of energy to break the bond.
So tl;dr: Nuclear binding energy is a subset of binding energy and yes, it is the same.