Severus Snape and the Boy Who Lived - Snape's Personality
The jumping-off point for this story wasn't really based on a change in Snape's backstory, but a change in his development from book one onward. When we think about Snape as a character, we're usually thinking about him as a whole - everything we know from his first appearance to his worst moments as a teacher to his final sacrifice. But in a lot of ways, this story was inspired by me looking at Snape in book one and thinking "He's not...actually...that bad???"
In book one, Snape is strict, he's biased against Harry, and he snaps at people. But snapping at people and taking points is the worst he really does. In the later books, we see him threaten a student's pet, refuse students medical treatment, insult students' appearances...he is undeniably an abusive teacher. In book one, he's mostly an excessively strict teacher, who's got a grudge against Harry.
So I looked at this, and I thought...well, Snape as a character has become more and more exaggerated over the books. And you can take that as a statement about Snape as a person, or you can say it's because of the author. And it would be interesting to look at Snape as if who we see in book one is the character that lasts.
Considering that I then immediately changed his book one actions, obviously I did something a little different here too.
My guiding principle for Snape, when I was writing Hogwarts Potion Professor, was that he was a teacher who cared about the safety of his students. His snapping at students who make a dangerous mistake, his hunting down a threat in the castle, his counter-curse on Harry's broom and choice to referee the next match...these all suggest someone for whom safety is important. And probably the most likeable characteristic of Snape in the later books is that he IS protective, and he IS trying to keep Harry safe. So I leaned into that, into him being a dedicated teacher focused on student safety. And many of the changes across HPP, in one way or another, actually are ripple effects from minor alterations.
So, coming back to your initial question, what changes in Snape's backstory and development could have led to this characterisation?
I don't think it would take much. Mainly, I think it would take the support of his colleagues as he adjusted to his life at Hogwarts, particularly Minerva and Filius. Snape is a relentlessly competent man. If he dedicated himself to competency as a teacher and a head of house, AND HAD PEOPLE AROUND HIM WHO VALUED THAT, then he could have developed in this direction. It's the bitterness of being constantly seen as mean, biased, and unreasonable that could make him so intractable.