Sorry, I guess I had ruled out the internal flash from all consideration.
I was not thinking of all possibilities. Yes, the internal flash is only a couple of inches from the lens, still farther and better than compact cameras, but yes, it can cause red eye.
The Red Eye Reduction tool delays a second while flashing the flash, attempting to cause the subjects eyes pupils to reduce in size, lessening the chance of red eye reflection (questionable if better). To eliminate red eye, move the flash farther from the lens, so the reflection angle coming back is larger, less likely to be seen.
There are good software tools to remove red eye in post processing, and they are good and easy.
Direct flash (like the internal flash) is very flat lighting, a deer in the headlights look (but proper exposure can help it). Off camera lighting, including bounce, is vastly higher quality light, more pleasing. The internal flash is very low power, and won't bounce.
It is something that you have to feel the need for first, but to improve your flash pictures, the answer for on-camera flash is bounce flash, which is a bigger flash, say like the SB-700.