Like most people (I assume), I thought the Pensieve was storing memories, but according to JKR it actually stores an accurate copy of reality. From an interview in July 2005:
MA: Do the memories stored in a Pensieve reflect reality or the views of the person they belong to?
JKR: It's reality. It's important that I have got that across, because Slughorn gave Dumbledore this pathetic cut-and-paste memory. He didn't want to give the real thing, and he very obviously patched it up and cobbled it together. So, what you remember is accurate in the Pensieve.
ES: I was dead wrong about that.
ES: I thought for sure that it was your interpretation of it. It didn't make sense to me to be able to examine your own thoughts from a third-person perspective. It almost feels like you'd be cheating because you'd always be able to look at things from someone else's point of view.
MA: So there are things in there that you haven't noticed personally, but you can go and see yourself?
JKR: Yes, and that's the magic of the Pensieve, that's what brings it alive.
So the Pensieve plays you an exact copy of the event you remembered, rather that how you remember it when you add it to the Pensieve. This means that when you watch it again, you’re seeing the same event exactly as it played out the first time, and not a slightly different version of it (based on your memory).
The two memories (the original and the watching) are based on the same event. When you watch it back, you might brush up on the details, rather than remember somebody different to the original.
The Pensieve recreates a moment for you, so you could go into your own memory and relive things that you didn't notice the time. It's somewhere in your head, which I'm sure it is, in all of our brains. I'm sure if you could access it, things that you don't know you remember are all in there somewhere.
So I don’t think it recreates the memory, so much as brings it to the foreground. As time passes, you might no longer be able to recall the specific details, but they’re still in there somewhere, and the Pensieve can bring them back up.
You might remember the new observations or thoughts that followed from rewatching the event, but I don’t think it would fundamentally change your memory of the event itself. For example, Dumbledore will remember finding Harry in his Pensieve, but he doesn’t suddenly have a copy of the memory which includes Harry as somebody who is “in” the courtroom.
I hope this makes some sense, and if not, I’ll tidy it up in the morning.