I'm not surprised by the lower back issues on front squats. That is a real killer unless you manage to stay up and down. Even if you do manage that you still end up using your lower back to support the weight. imagine putting a weight on your back and leaning back a bit, your abs will engage like hell to stabilize you. Same thing happens on front squats if you lean forward. Your lower back is engaging to keep you from nose diving. Looking at your videos I would say that you are not keeping your back arched. You can see the top of your back rounding out which is going to move the weight forward. Can't see your elbows, but I would guess that they are dropping when you are driving out of the hole. With your elbows high and your back tight it shouldn't be an issue.