I would confirm that, to me, a great actor is someone who makes their character believable, to the point where I see only the character, and not the person playing the part.
Some actors are mysteriously lauded for their utterly one-dimensional 'abilities' - the pinnacle being Sean Connery, who is only ever Sean Connery, and played an Irish cop with his un-losable Scots accent, and got an Oscar for it!
As advised, John Wayne is a similar example, as is James Nesbit, Michael Caine, the list goes on.
Watch someone like Alan Cumming in The Good Wife, and I can completely believe he is an American, born and bred, even though I know he is as Scots as shortbread.