In my understanding none of the two stretching methods is "linear" because the influence is different on different parts of the histogram no matter which one you use for stretching. If you had a linear image before it will be non-linear after the stretching because it is the aim of the stretching to bring out contrast and detail and brightness mostly in the dim parts ofthe image while not affecting the lights.
Stretching with the "levels" tool (ctrl-L) is fine in the beginning to "stretch" only the shadows without affecting the lights. In the histogram this results in a broader "background peak" It is more a "pre-stretching" for me that I do in the very beginning of the precessing to prepare the image for gradient remocal etc. and further stretching with the more flexible curves tool. Leave the right slider alone during the whole process. First set the black eyedropper to 25,25,25. Then shift the mid slider iteratively in small steps to the left. After some iterations a gap appears at the left side and I click with the black eyedropper into the darkest part of the image to close the gap and set the background to 25,25,25 again. I have created a small action that sets the mid slider to value 1.1 to let me do the iterations fast with a shortcut.
Later in the stretching process I mostly use the levels tool only for setting back the background to the predefined value with the black eyedropper from time to time as a kind of normalization..
Later in the processing workflow I mainly use the curves tool (ctrl-M). Also here it is important to lift up the curve in tiny iterative steps. In difference to Scotty's second demo-picture it is also important to keep the right part of the curve
always and strictly on the "base line" by clicking at it and bringing it back with the arrow keys before you hit enter. If you do not do this you will ruin the stars. It is good to always fix some "anchor points" on the curve left and right of the curve part you want to stretch to prevent the lights and the shadows left and right.
Again I strongly recommend Scott Rosens tutorial which is a masterclass help for understanding what how stretching is done in Photoshop.
P.S. Sardinia skies are also still cloudy
Cheers to all out there also waiting for clear and dark nights!
Pete