It is often frustrating in Illustrator to bring up the gradient dialog, select one of the colors in the gradient (so the triangle above turns black), and then try to pick a color you already have somewhere else. Illustrator will replace your entire gradient with the solid color you select. There are three ways to get around this problem:
use the color selection tools that show up when you double click the color stop
make a swatch of the colors you need, then after clicking the color stop, hold Alt and click on the swatch
after clicking the color stop, press “i” to enter eyedropper mode, then hold shift and click on any color - this can be in your artwork, or anywhere on any screen if you click and drag away from Illustrator