What we want: Some kind of "physical" unit of measurement, where 255=100% means maximum number of photons, 128=50% means 50% of photons, 64=25% means 25% photons. This means gamma=1, linear color space, calculations are easy and correct in this unit.
What we have: Eyes that need ~4 times the light to see something twice as bright, that means a gamma of ~2. 255=100% is still white, but 128=50% means ~25% of photons that LOOK like half a s bright. It is a really good idea to store pictures with such a gamma. Monitors were adapted to this and also use such a gamma, 2.2 is common today, once for good technical reasons, today maybe more because of tradition and compatibility, don't know...
If you want to blend black with 50% white and expect a gray that LOOKS like average brightness, then you will like gamma ~2. But that also gives you the dark blends with colors.
If you prefer a gray of 50% photons that looks much too bright, then you will prefer gamma=1. But on the other hand this looks good with colors.
To me it looks most useful to keep simple formulas and just store the data with the desired gamma value.
Conversion between gammas are usual. Cameras have gamma=1 for most of their range, so gamma is usually converted to ~2 when a Raw is turned into a Jpg. If you would just send a picture with gamma=1 to a monitor that assumes gamma~2, the result will look much too dark. If you work with gamma=1, then the conversion will be done when the final result is sent to the screen. That's where color management starts... Profiles can handle different gamma values better than a formula in a program.
-Check if you have color management enabled, "edit > options..."
-Check in the attributes if your picture comes with a correct profile. If it comes without a profile, assume that it is sRGB and assign sRGB using "tool > colors > set color profile". Or just double click on the profile part in the attributes.
-Enable 32Bit for the document or all the layers. You do not want to have a linear color space in 8 Bit.
-Convert the whole document to a color space with gamma=1 with "tool > color > convert with profile...". All opaque parts of the picture will still be the same, but the blending will give different results.
I hope I did not simplify this receipt too much. PL can have profiles for each layer, and when you are in document mode, the layers are converted to the right space on the fly, but imho that's already a different topic.
You will have different results if you use proPhoto or sRGB. That's not because of different calculations, it's more because of the rendering intend when the result is sent to the screen.
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It's somewhat mindbending to see results assuming different gammas. What is the gamma of the data, what gamma did you or a profile use to correct it?
Your test picture comes without a profile, so I assume it's sRGB. That means the data is stored with gamma 2.2. The topmost area is 50%, so this is the average of data with gamma 2.2.
Now if I convert the picture to a linear sRGB using the receipt above, the bottom area shows values of 50%. So this is the average with a gamma of 1.
