The definitions you cite for "on course" are unclear because it's not a command or clearance it's a statement of ones positon. I always use either "procede on course" or "cleared on course". Secondly, to address your question, does not imply altitude clearance. If you look up "course" in the PCG you get:
a. The intended direction of flight in the horizontal plane measured in degrees from north.
My emphasis added.
The definition for "resume own navigation" is a little bit more ambiguous, but i've only used it for lateral course guidance. As always it's best to clarify if unsure. I use "resume own navigation" for an aircraft that isn't going direct somewhere ( e.g. doing airwork, on a sightseeing tour, etc.) and follow it up with an altitude instruction.