The backstory they were going with looked roughly analogous to what I recall from the comics.
Except you're never quite sure in the comics if it's his actual backstory, if someone's screwing with him (again), or if the author's screwing with us. E.g. look how his daughter came into the picture - from him sleeping with her mother in a jokey retcon of him in a 70s blaxploitation Cage/Fist story.
Maybe personal preference but I liked the backstory where Wade Wilson was just a name he stole from a merc he killed.
Of course if I were in charge of it the backstory would be an A-Team style voiceover - "In 1990 a popular villain from another company's teen team mag got even more popular by showing up in a certain pointy eared character's books. So the most '90s author ever decided to palette swap the exact same character but change the name very slightly. the 'new' character then got really popular despite writing of sporadic quality and an awful movie version that we're still very, very sorry for. And now, Sessions presents...Deadpool.'