GOD OKAY SO i skimmed through the script specifically to look for discrepancies in the character names because i had an inkling they would do something like that and i found exactly what i wanted. i formally present my findings:
so for wade the distinction is pretty clear cut; when the suit and mask are on, he's referred to as deadpool. when he's out of the suit, or even just when his mask is off, he's wade.
pretty simple even though it IS still interesting--he's literally masking when he's in the suit; he adopts a persona that comes off with the mask. but here's the fun part! LOGAN doesn't wear a mask throughout the whole movie, and yet he is referred to as the wolverine sometimes and logan in others.
firstly, take the bar scene:
he's introduced in the script as LOGAN, but the moment the bartender attacks him and wade questions if he of all people is going to let himself be spoken to this way, he becomes WOLVERINE. a defensive wall. and it switches right back to logan the moment he says "you don't want this"--back to the person he is who's lost everything, who doesn't want to fight anymore, instead of his x-man mantle.
same with the scene in the diner:
he's wolverine until he finds the rubbing alcohol and downs it. then he's logan until wade brings up his position as an x-man and the version of him who died, and then he switches back to being the wolverine.
and one part i find especially intriguing is the scene in logan's mindscape:
"cassandra and LOGAN stand in a gorgeous, ethereal place." "WOLVERINE falls to his knees."
i found it very curious that when he confesses what he did to laura, he's logan all the way through, but he's wolverine when talking to cassandra about the same thing, even though she did have an effect on him.
you could argue that he has his walls up because he's still wary of her, or that it's ironically even harder for him to talk about the past when he's in his own mind, or that he was aware of what was happening outside the whole time and part of him had been playing into it on purpose even through it all. i think all of them make sense in their own way.
there are a few more instances of this, but tldr i just think it's so so fucking cool that they play with his two different identities in the dialogue names to show when his walls are up. it's a fantastic visual representation of which side of him he shows at what time, and it's just great direction too for the actors reading the script.
he's wolverine throughout the entire honda odyssey fight scene and even in the hideout, or when he's reminded of who he thinks he's supposed to be and, consequently, of his failures. he's logan when he tells laura about what he did. when he's stripped down to his most vulnerable, or when he can forget about being the wolverine for a while, when drinking helps him forget--
or when he's back home with wade and has found peace within himself.