that's fine but Travis Green's opinion is beside the point, he is not in charge of what definitions are. Unless he splits his talent evenly across all his lines (which he won't) and gives all his lines equal icetime and equal offensive faceoff time (which he won't) then he will have a 3rd line and fans will be correct in referring to it that way. And I'm confident Green himself will use these phrases at some point.
I'm not the official authority but pretty sure "who you have him playing with" and "what situations he is going to be used in" are precisely what define 1st/2nd/3rd liners. And 3rd liner indicates players who have less opportunity for statistical production, in a league that rewards money based on statistical production, so I'm a bit surprised that fans would argue this isn't relevant to a player.