I would not use "teach on course" in that meaning.
Student or Learner
I'd say 'teaching a course', 'teaching this course'... sound natural, but sometimes I do see people say/write 'teach on a/this/the... course'. Are both versions correct and natural in English? Thanks
I have never heard "teach on a course" as an equivalent to "teach a course.". I consider it incorrect. You can teach something on a course. For example "I teach calculus on so-and-so's Introduction to Mathematics and Science course."
Unless, of course (n.p.i.), you're a golf pro.
That non-pun is sub-par.