Leg presses can give added support to the lower back because of the brace effect of the seat your sitting on. With squats, not so much. Though front squats may offer less lower back stress than regular squats. As any exercise, where the bar is resting directly on the shoulders/neck/traps , the spine (in direct line) will be affected. Some find less lower back stress, when doing leg presses, by doing one leg at a time. Not too sure what you mean by "mat' when doing leg presses. You laying on a floor mat; under a old style machine, bar placed on your feet, etc...? Coming down 90 degrees is not the best of ideas. Not too many people do. Try breaking a bit before that.
Many times a lower back will need a proper warmup. Most lower backs lack hardly any range of flexibility, being stiff with limited motion. This can cause serious injury, sooner or later. Something to consider with a bad, or potentially bad, lower back. Keeping the lower back warm doing exercise is a good idea.
Didn't say if you ever used a Hack machine or have done BB Hacks. Less stress seems to be gained with the weight (BB) held below the waist/near the center of the body. Only exception would be the DL, where leaning forward, into the weight, can also put unneeded focus on the lower back.
Might get the back checked, just to clear the mind, one way or another, if you do have a ruptured disc. Good Luck.