Or any other advanced word in that category.
If this is sbout putting a $10 word in a college essay, rethink.
Imo, neither word works without context. They describe individuals. Revisit your dictionaries, really, for the usage.
imo, the best word to describe a surgeon should always be steady-handed
Arrogant? 
At least, you need a high degree of self confidence to cut into the body of another human. But I don’t think I’d put either of those into an essay. But remember this. If you are applying to undergrad programs, they don’t care if you have the qualities to be a surgeon. In fact, students who are SO SURE of their career path can be a slight turnoff to admissions. They see college as a place to explore and learn, and maybe discover new areas you are interested in. Don’t think every breath and action has to be pointing to that career goal 10 years down the road. They also know how many students who are targeting med school don’t end up getting in or change their minds (lots and lots). So percentage wise, your odds of making it to being a surgeon are pretty small, and they are well aware of that. Bottom line is that describing yourself in terms of what you hope define a future surgeon isn’t very useful. They’d rather have someone curious, engaged, or interested, I think.
Noone knows what those words mean. Dont use them.
That’s an awfully dim view of admissions officers you have there.
And of us posters. 
Neither word is quite correct. I would opt for “meticulous” but I really like #2 “steady-handed.” How about cool, calm, and collected? That’s what I would want of my surgeon. Not sure I’d go with “The Good Doctor,” but that might drive this thread right off the rails.