You are using flash (often referred to as "strobe"). These lights emit a strong pulse of light that lasts for a VERY short time. On some high end strobes it may be less than 1/10000 second. That's the key thing you haven't grasped, I think.
You fire a shot at 1/200 second, and the pulse happens during that time, and the image is correctly exposed. You change your exposure to 1/60 second, and the pulse happens during that time, and the image is correctly exposed, and exactly the same as 1/200 second. The extra time the shutter was open makes no difference (well, the background light gets more time, but that's usually negligible by comparison to the vast amount of light from the pulse).
That's why the shutter speed is (almost) irrelevant. Does that help you understand? That's why you can only control flash exposure using aperture and ISO.
Now why did I say "almost" irrelevant? There are two reasons:
1. background light - if there's enough background light, the shutter speed will affect how much of that you see in the picture. Have a Google for "dragging the shutter" and "second curtain sync".
2. flash sync and shutter curtains - to take the picture properly, the entire sensor must be revealed when the flash pulse happens. The shutter (on lots of cameras) consists of two curtains, and the shutter speed is a measure of how far apart the two curtains are. At slowish shutter speeds, the first curtain opens, the entire sensor is revealed (flash!), then the second curtain closes, and the image is captured. At higher speeds, the curtains travel some small distance apart, and there is no time at which the entire sensor is revealed - what is revealed is a moving band across the sensor. If you try talking a flash picture with, say, 1/2000 second shutter speed, you'll find a sliver of the image properly exposed, and the rest dark - the sliver will be the portion of the sensor that was revealed when the flash pulse arrived. That's why each camera has a quoted "flash sync speed" - it is the fastest shutter speed for which the entire sensor is revealed at once - any faster and you'll start to see the "sliver" effect.
That's not the greatest explanation ever, but at least you can go Googling for sync speed, and shutter curtains, and find out the rest.
Hope that helps.
they say you have to take a thousand bad photos before you can take a good one - I have mastered the bad photos part...