I'll start with you: It's a little hard to see since you're wearing dark clothes, with dark tack, on a dark horse, but you look generally good. As you said, your elbows could be bent a little and then follow Luc's neck movement. I like that your eyes are up and you're looking where you're going.
And Luc: He's looking very nice for just 3 years old! He's trotting along comfortably and seems to be listening to you. I would encourage him to really engage his hind end and keep up his tempo. In the first Canter video, he broke into a trot a couple times. Be ready to encourage him forward before he breaks gait (especially in the corners). Also, after he broke into trot, and you asked for the canter again, he seemed to kind of rush into it. Since you said you're working on transitions, might as well make each one as perfect as possible (even if he's the one who is causing the need for a transition!)
And my biggest piece of advice...wear a helmet! You're on a 3 year old! You never know when he might do something to accidentally dump you off. Like, for example, jump half-sideways over a blue barrel on the edge of the arena,,,