I hesitate to get involved here, because with so man different opinions, and one poster apparently proving the adage that 'A little knowledge is a dangerous thing' but your brush folder is a mess, and I'd be very much inclined to simplify its contents.
1) Delete all of the brush folders, and add the Legacy brushes (brosses héritées)

It is possible to twirl open the Legacy brushes, and then the Default brushes and drag the brush presets, one at a time, to the area below the folders. It is a bit of a pain that you have to do this brush by brush.

When you have moved all the brushes, you can twirl shut the Legacy brushes folder, and drag it below the brush presets you moved, so it looks like this. You would then have something close to your first screen shot.

You could even delete the folder leaving you just the presets.

At any time, you can reload the Legacy brushes folder, and all of its presets will be there.

I personally find the new system far too cluttered. It can be difficult finding the presets you want from the large number of of folders and presets, so I try and keep it simple. So you could create a new folder with your most used presets by dragging them from other folders, or making new presets yourself.

Then decide how much information you want on the screen. I prefer to keep this to a minimum because I know what brushes I use most of the time, and I want to keep the screen clean and free of clutter.

I hope this helps. I also hope that the next major update of Photoshop improves the way brush management works, as it definitely needs some work as it stands.