I've had some musicians state that the following tuning is Drop C tuning (E is dropped to D, and then all strings are dropped 1 additional step)
That's generally what is used, as it the same relative tuning as drop D, so from a chord perspective, the fingering is the same, only at a lower pitch. The first two strings make 5ths which makes it possible to play chords more like fingering a riff, like Outshined. Especially in hard rock, there's been a history of dropping tuning to achieve a deeper sound, first with all the guitars tuned down 1/2 step and drop D tuning, to drop D, only down one half step, then down another half step to drop C. Chevelle's Face on the Floor, nearly all Sick Puppies and many others use this tuning.
If you follow the Drop_C_tuning link, you'll see that the majority of those listed are using CGCFAD.
I've also had other musicians that state the following tuning is Drop C (only E string is dropped to C)
Playing with a major 6th between strings would be awkward, even on a bass, but, the terminology could certainly be used for both, since it is a short form.
5-string bass players can be a bit more flexible, using the same tuning when accompanying players using standard or drop tunings. Also, I've seen one bass player tune in fifths, like a cello (CGDA) for more range on 4 strings.