How can I get a less extreme, more controllable result than Auto Equalize in GIMP or ImageMagick?

Gimp's Equalize command automatically adjusts the brightness of each color so that the histogram for each is as flat as possible. If you are unsatisfied with the results, you can do the same manually and then follow up the brightness adjustments with saturation adjustments to each color channel.

Note that these are extreme adjustments to a small source image in a lossy compressed format, and pretty much any results will look bad. Even the dust spot on the sensor at the bottom is exaggerated. It's a lovely photo but a bad choice to test this method on.

First, open the image in Gimp and select Colors > Levels.... Working on one color channel at a time, move the input level adjustment sliders so that they very closely match the spikes where most of the color is represented. It will look more and more like your example as you go.

Next, select Colors > Hue-Saturation.... This is a little trickier but follows the same principle. Choosing one primary color at a time, de-saturate the value until you are satisfied with the result. In the above example I had to adjust both the Red and Magenta values to affect the 'redness' of the image. You don't necessarily want to go all the way to -100 for every color or you will have a greyscale image. Just reduce the saturation so that it more closely matches the source image.

Once again, this is an extreme adjustment to a compressed image file format. The output will never look very good. Best practice would be to shoot in camera raw format, use a source image format with greater bit depth (e.g. 16-bit TIFF), or work with a set of bracketed images to fuse together.