These are all about usages. I've never heard a top ski instructor or pro or racer use the word "expert" with any great respect. That's a word that seems to be used more often by people who aren't.
Pro tends to be used more often in the sense of highly trained and skilled rather than one who gets paid.
I'd rather be a pro than an expert. Also I'd rather be an instructor than either of them. It's just a word that I have felt was attached to my profession since I was a boy. This doesn't mean it's a good descriptive word.
Probably the best word in the teaching realm is coach. But historically that's always been attached to "race coach" for me.
Ultimately the nicest thing you can say to me, or the nicest thing I can say about someone else is, "He's a good teacher." Or "She's a good skier." I've never had anyone ask me for a certified pro or instructor, but many who have asked me if I knew a good instructor. (I even had that on my biz card once: WW, Real Good Instructor--instead of certified, etc.)
In my hierarchy of skiers, I see Beginner, Intermediate, Expert, and Good.
Again, it's more about usage (and who's doing the using) than definition.