Yes, honors classes are limited to honors students.
I don’t think regular courses become “honors courses” through additional work.
What you may mean is that sometimes there are two different courses/course numbers for a particular class, and one is honors.
For example:
There is regular Philosophy Deductive Logic and there’s Philosophy Honors Deductive Logic and the main difference was that the Honors version has one more exam (which I don’t remember if that extra test covers an additional module or what). My kids took the honors version and it was a “work at your own pace” class where typically if you work hard, you can complete the entire class by late Oct/early November. There was no final exam.