others with a more developed press will chime in but the first thing that comes to mind WRT strict press is your back is arched and your pelvis rotated back. (anterior tilt) this mean that when you reach the sticking point, you're bending back at the waist putting shear force on the lumbar spine...you will feel a lot of static compressive force but that's what the spine is pretty good at.
take a slightly wider stance and focus on squeezing your ass cheeks as hard as you can at the beginning of the press. keep your legs rigid and knees locked or close to locked. focus on keeping a level pelvis if you think of creating posterior tilt, you'll probably have it. Think of a hard deadlift or kb swing lockout....I'll see if I can find good article on the press to link over.
push press is of course slightly different, an Oly guy can help you there.