Could be several things, though both screens all at once would speak more to the electrical world than delamination of the polarising layers in the LCD screens (would usually expect to see that creeping before total, and one screen at a time). I don't know that I have seen this specifically for the DS lite before either in general or a failure in some other repair/disassembly, but it is about the time (probably started being made in 2005, to be sold in early 2006) we start to expect to see the weird and wonderful failures. Have seen similar things in other devices over the years though.
Pushing can change two main things.
1) Some kind of ribbon cable being loose, though each screen is independent and on opposite sides of the board which makes this questionable at some level. I would not expect force to repair two cracked ribbons at once.
2) It flexes a board* which makes a bad connection good, though again both screens at once is odd.
*it is not impossible that a warped case causes issues.
If you can find another working DS lite you might try transplanting the board. This might be easier said than done though (days of cheap DS lites seem to be over, maybe you get lucky though).
More likely then you get to remove the panels and see what goes when the board is exposed (possibly even by itself if you are really bored).
Obvious things to look at would be the ribbon connectors (they might be close enough) and pathways from the video related chips to them (basically what is connected to the ribbon connectors).
Desaturating while backlights remain on would speak to something after it gets turned into the slightly more analogue signal the screens take in being troubled.
Hopefully it is some leg on the chip or some resistor to it, maybe connector, that suffered a bit of a creep failure such that you can reflow it and carry on with life (if you have a hot air station or IR setup available to you then might even consider a reflow, possibly even just the likely trouble areas, just as a course of action -- if you have the board out already then it is a quick and easy process with minimal risk if you know what you are doing*).
*probably not dealing with crazy precision stuff here so amounts to be careful of plastic things, don't burn it and don't have the air up so high that it blows your components off the board when the solder goes liquid.
Looks like a reasonable video on pulling it apart which is also where you will find the ribbons in question to reseat if you wanted to start there first.
