I think logic will tell you what the problem is:

When all three modes are used without flash, the image looks the same. When all three modes are used with flash, the images are different.

So, let's investigate the flash.

Now, what makes a flash work?

Well, we have a power source, a xeon bulb, and a large capacitor.

More than likely, your batteries didn't die in between shots or deplete enough for that to be a problem, so let's rule that out.

This leaves the xeon bulb and the capacitor.

Well...the xeon bulb is expected to last a very long time and if it is fired with the same voltage constantly, it should provide very constant results. If it didn't then no one would purchase electronic flashes as they'd be unreliable.

That means it is could be the capacitor. Now, you saw the light telling you that the flash was ready before you fired; however, just because the light is on, that doesn't tell you what the charge of the capacitor is. I'm more than willing to bet that there were different levels in the capacitors each time it was fired.

If you could, try repeating this expierment, but change the order: A, S, P or some other variation and compare those pictures.

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It could be that or...in P mode two lights aren't on. One on the building and one on a lamp post.