I think this would be misleading once you dive deeper into particle physics. The strong interaction is really »the interaction mediated by gluons between color-charged things«.
• Gluons interact with gluons, without the need for quarks.
• Many (almost all) bound quark states are not found in nuclei, only uud (protons) and udd (neutrons) are. But there are also all the mesons (e.g. the pion), and a whole lot of other baryons (xis and sigmas and what have you) exist.
To put this into perspective, it feels a bit like calling electromagnetic interaction the »chemical interaction«, because chemistry is explained for the most part by the interaction of electrons. But that would leave out a lot of different ways matter can interact, like Bremsstrahlung, positrons, proton/proton repulsion, and all that.
It's something you never get used to. As you get older, this will just keep happening. We used to put commas before the last item in a list back in like the stone ages when I was in school. My SAT score looked really lame for a bit of time when those suddenly changed.
Is there a reason we're leaving "nuclear" off these forces' names now?