You missed my point. Your not adding new solid color fill adj layers to change color, you are double clicking the solid color fill adj layer to change the color. You can however use a solid color fill adj layer clipped to each part of the image you want to colorize.............again , setting them to multiply.
You can stack/clip new color fill adj layers if you wish, you would just have to turn the one you are not using off.
When you stack color fill adj layers and you use "color" you get a mixture if you don't turn off the previous layer.
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This is what it would look like if set to "color" and the first color fill adj layer is turned off. Does not work with color blend mode.
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This is what it should look like set to multiply and one color fill adj layer turned off....
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However, I was a bit confusing when I spoke of the foreground and background color picker and I have now edited that above! I wrote this because the OP shows a color pallet on layer 4 in his video. I was explaining a method of sampling those colors in his layer 4 with the foreground and background color picker, then transferring it to the solid color fill adj layer.