There are some substantial differences between the scenes in this movie and Wolverine's recollection of the event as seen in X-Men and X2. For instance, in X2, Wolverine remembers blood gushing from his hands when the blades slide out for the first time, and pushing a door open on his way out. In X-Men Origins: Wolverine, there is no blood coming from his hands, and he cuts through a wall on the way out. Also, in X-Men, Wolverine remembers being cut open with a scalpel, while no such device is used in X-Men Origins: Wolverine. The real reason for these continuity breaks is, of course, that a different team of director/writers was responsible for the latter film, who gave their own interpretation of the events. Even the room itself (which consists of two levels instead of one) and the basin in which the procedure is done are quite different. On the DVD, the production team explains that they were not satisfied with the design as it appeared in X-Men and X2, and deliberately changed it. Maybe the room itself has seen some redecorating over the years, following Wolverine's rampage, but this seems somewhat doubtful, as Stryker and his team prepare to permanently abandon the facility after the procedure. So there is no real narrative explanation to the room discrepancy, and it should therefore be considered a retcon.
The same can be said from Wolverine's different recollection of the course of events, the screenwriters simply made up their version of this event. However, there is a somewhat plausible narrative explanation for this. As any psychologist can relate, eyewitness reports from memory can be very unreliable. This is because events are not remembered as a single unit in one region of the brain, but as fragments stored in separate parts (e.g. hair colors are stored in a different part than sounds and faces). Discrepancies can occur when people try to retrieve the memory; details get mixed up easily, and elements of different stories are inadvertently combined into one memory. Another complicating factor is that Wolverine's memory has been violently altered, leaving him with only fragments and images of the procedure, so it is in fact very easy for him to have a flawed recollection of the real event. Not to mention, when he remembers he is in fact, having nightmares. So his bleeding, being in pain, being sliced open, could all just be part of the dream, muddling with what really happened.