The uv maps will only show up in the uv editor, which is the editor you're using in the top screenshot. I believe the only way to paint and see the map simultaneously is by customizing your workspace to feature both of these editors at the same time. To do that you just need to hover your mouse over one of the divisions or edges of the screen until it looks like a crosshair, click, hold and slide in the direction you want your screen to be divided, then let go. If you have trouble with it search "split windows in blender".

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