To be honest, it's a bit hard to judge from those videos - it's only one angle, fairly close up, etc. From what I can see, you're maintaining a nice lumbar arch and have the basic idea of the lift down. As others mentioned, don't shrug the weight, at all. As for the leg/back interaction and transition, it's worth noting that the ONLY role of your back muscles in the deadlift is keeping your spine rigid against the load. The only thing the muscles around your spine (erectors) should be doing is maintaining that nice lumbar arch - they should NOT be lifting/pulling the weight to straighten you out. You can basically break the deadlift down into two stages, although this is just a mental breakdown - during the actual lift, it should all be one smooth motion:
In the first stage, during which the bar moves from the floor to (roughly) your knees, the weight is moving because you are straightening out your legs (largely a quad movement). During this time, your back is basically maintaining a constant angle with the floor, which means all your back muscles are doing is keeping your spine rigid.
In the second stage, once the bar clears your knees, you need to straighten your body out. Although your back will now change its angle with respect to the floor, the main mover will NOT be your back muscles - they are still just doing their main job of keeping your spine rigid. The actual motion should come from you pushing your hips forward with a very hard contraction of your glutes. You should actually feel your butt muscles tightening up a lot to drive your hips forward. Not only does tightening your glutes this way keep your back safe, it also lets you incorporate your hamstrings and generate a lot of power.
So, to recap: during the entire lift, your back muscles are *only* working to keep your spine rigid (which is hard enough), while your legs/glutes/hips/hamstrings (in succession) are doing the actual moving of the weight and your body. Just be sure to keep the bar in contact with your legs the entire time, as it's a *much* more efficient (and safer) motion if you literally drag the bar up your shins, over your knees and up the thighs. No, it's not particularly pleasant, but if you don't have scrapes and bruises up the entire length of your leg after a deadlift session, you're not doing it right.