Either could be used and neither would be wrong. The meanings are similar, but there may be some slight differences.
Normally, I would use on. When I am word processing, or checking email, or working on a spreadsheet, I am usually inclined to say:
I am working on my laptop.
Typically, on is the preposition I'd choose.
However, there are cases where I might change that. Let's say my coworker Linda owns a computer at home, and I know for a fact that her hard disk crashed two days ago. (She's also borrowed a company laptop, but I'm not aware of that yet.) At 10 o'clock, I notice Linda's desk at work is still empty, so I ask Gary about this:
J.R.: "Gary, do you know where Linda is?"
Gary: "Yes, she's working from home today."
J.R.: "Wait – how can that be? She got her computer fixed already?"
Gary: "No. The folks in I.T. loaned her a laptop yesterday."
J.R.: "Ah, I see. She's working with a laptop."
This works, because the phrase working with can be used to mean using. As another example, if I was in a wood shop, I might say:
I'll be working with hand tools today.
if I'm not planning to use any power tools.
This is another one of those cases where prepositions are simply too flexible to say there is always one correct preposition to use.
The first is fairly common when speaking about someone's circumstances, such as their financial state, their job title, or their history. Basically it refers to being in someone's shoes, so to speak.
He's buying a brand new car with all those debts he has? I wouldn't do
that if I were in his position.
The second isn't common at all and the only way it works is if you're talking about someone's physical location. I can't even think of a good example for it.
John's on the US/Canadian border and he's going back into the US? I
wouldn't do that if I were at his position.
Best Answer
The form I have encountered is:- He plays games on his laptop. I don't particularly know why - it just seems idiomatic.
Of the three versions you have suggested, the first sounds wrong, as it suggests he is inside his computer. I would argue that the second and third are both correct, though the third has a subtle difference in meaning.