Neb, the red eye tool is just like some of the other auto adjust tools in
PS. On some images they guess well, on others they make things worse. The red eye tool does seem to be somewhat particular to the size of the marquee which you draw. If it is too small, it will correct only part of the red area. If it is too large, it tends to offset where it makes its circle of correction. I recommend that you experiment with the size of area that you select. Sometimes you need to go considerably larger than the red area.
There are many ways to remove red eye in
PS. Butch's method is both popular and effective. Other methods involve desaturation of the red but this often leaves unnatural looking "black holes".
For images where the iris is still distinguishable around the pupil restoring the color is a preferred method. This can be done by painting or selective color adjustments.
When brown eyes are involved, I like to remove the red and restore the brown which is a bit tricky since red is a key component of brown. For that task I find the Color Mixer is very useful. Attached is an example. You use the eliptical marquee to select the eye areas and do Image>Adjust>Channel Mixer. With the default channel on Red, reduce the red slider to 0; the green slider to 80; the blue slider to 20; and the Constant Slider at the bottom to +20.
Regards, Murray