Overwriting a brush preset without creating a new preset?

I'm a fairly new user of Photoshop specifically, coming from using other drawing applications.  I can't figure out how to overwrite a brush's settings, so as to save it in-place as the same brush I was using, name and all, but updated with my new changes to it. 

What I gather is that you click "Create New Brush" at the bottom of the Brushes window, or "New Brush Preset" from the triple bar icon's dropdown at the top of that window, and save yourself a new brush with a new name, that includes the changes you've made to your preset.  And now you have a new brush that is in many ways identical to the brush you were using, and since all you really wanted to do was *slightly change* the brush in the first place (e.g., lower the scale of the texture), you now have to delete the first brush, since you no longer need it and it'll just end up creating visual clutter in the brush group? 

I could be missing something really obvious, but I have a feeling this is just how it's done(?).