The bar looks to be moving pretty straight up your legs over mid-foot, but you're right that you're not really squeezing your back into extension before pulling. We often cue "squeeze chest up" for this, as the part of your body you're most aware of and can see, that moves, is your chest. But it's actually your back muscles, primarily the spinal erectors, that get into extension and via your straight arms, pull the slack out of the bar before lifting it. So, do that.
As far as your grip, I do like the general idea of doing some DOH work, but it's also preferable not to introduce something new and different at the working weight. Do your last warm-up set with the same grip you're going to use for your work-set in the deadlift.