I think my response to this is fairly simple.
I am writing a Mass Effect and Falliout 3 novelisation (actually I've finished ME1 and am working on ME2). I have read three other FO3 novelisations and two ME2 novelisations, and not one of them, including my own, tells the same story.
I guess you could argue that it only applies to Role Playing Games but I don't think that's true either. Even taking the most simply experience, say of Lara Croft fighting a boss, and ask a whole range of gamers how it happened.
If you read the Lara Croft comic book and asked a whole bunch of people how a fight happened, they would all basically tell it the same way, because the source is the same. 'Lara did this, then she did this, then the boss died, the end..' And no matter how many times they read the fight scene, or how many times you asked them to describe it, you'd always get the same sequence of events happening the same way.
Now ask someone to explain how they killed a boss in a Tomb Raider games. Some will describe a daring and terrifying drawn out firefight, with Lara hiding behind any cover she could find, bleeding and wounded and with no way to heal herself. Others will describe a boring long range pot-shot match where Lara covered herself in bandages whenever she got so much as a splinter. And the even better part is, if you asked the same gamer to play it twice and describe his experiences both times, you'd get an entirely different experience just from one person.
It kind of comes down to characterisation through the act of playing. Gamers went ballistic over the changes to Samus in Other M, and part of the argument was always 'just because she never had a voice didn't mean she wasn't characterised.' I think the Extra Credits guys even did a whole video on the things you know about Samus simply through gameplay. Well if everyone plays the game differently, everyone will have adifferent view of who and what Samus is, ie. They will have created their own character. Back to Lara Croft because I don't actually play Metroid.
I tend to play Tomb Raider very cautiously. I don't use healing items often and quite regularly find myself limping through really tough fights with nothing more than my pistols because I don't like wasting ammo either. My view of Lara is thus of a tough survivalist who understands that you save your insurance policies for when you need them. My Lara Croft is a very mobile fighter who trusts in speed and agility over a big gun and determination. She hoards everything, not just ancient relics. She plots her course through platforming sections before she begins, and dislikes when she is rushed into things.
Someone else playing Lara might have her as a reckless character who makes leaps of faith more often than plotting her course, who uses all her ammo for her biggest gun on a little bunny rabbit. Yahtzee made a point about how he views Lara as a remorseless career thief and apart from some dialogue during the cutscenes, or some backstory that gets revealed in the guidebooks etc there's nothing that says my interpretation of her as an altruist who doesn't like killing (a view I took from reading the comics) is any better or worse than his.
I guess to try and pull this back on point is to say that if you read a book, beyond speculating about a character's motivations if and only if they aren't spelled out, the character of a novel or film will always be exactly the same. It cannot be changed simply by force of will, once something has been set in stone about the character, it will always be that way. The Lone Wanderer, Samus, Lara, Shepard, any character you care to mention, their actions are informed by the way you play the game, and thus their personality is always partly under your control, because no one will ever play the game in the same way that you will. They won't, to compare to books again, read the same words as you do. They are creating through their actions.