The Latin catulus did indeed mean "little dog" (i.e. puppy). Whether it is the origin of our word cat is more difficult.
Some writers on etymology assert that the Old Germanic languages took the word which has come to us as cat from the Late Latin cattus, in turn derived from catulus. You'll note that most other European languages did - on the face of it - similarly, rather than take their word for a cat from the Classical Latin felis.
Not all the evidence which would be needed to prove this notion exists, though. The Latin cattusmight have come from Celtic, or from a North African language (that the Arabic is qitt is suggestive here) and its similarity to catulus could be no more than coincidence.
The jury is to some extent out, as with so many etymologies. But that presented in C10 is one of the common theories.
(I don't have time to give proper sources right now; I'll come back to it another time.)