Is there a difference between the strong nuclear force, and the strong force (without the nuclear in between)? I have heard that the strong nuclear force binds protons and neutrons together, while the strong force binds the gluons together inside protons and neutrons. Is that true?
At a fundamental level, there is only one strong force, which is mediated between gluons between quarks and gluons with non-zero color charges.
In practice, due to confinement, at large distances (or equivalently, low energies) compared to the confinement scale, there are are no objects with a net-color charge. Loosely speaking, this scale is some fraction of the size of a nucleon.
Therefore, inside a nucleon, free quarks and gluons directly interact via the color force.
Nucleons can also interact. In the low energy effective field theory description of nucleons, the relevant degrees of freedom are nucleons and pions, which are color-neutral objects. Nucleons attract each other by exchanging pions.
However, the ultimate, fundamental origin of these nucleon-pion interactions is the same strong force that binds quarks and gluons. The effective low energy description is a convenient way of describing a process that, in more fundamental terms, would involve many complicated interactions of quarks and gluons inside the nucleons.