You two make a beautiful pair! I definitely see potential for English. And for only riding English less than a month... wow!
You have nice leg position. The only thing I really see is that your hands are too far apart. Think about keeping your thumbs to the top... otherwise, good work. I like how you have soft, "giving" hands.
I noticed though you kind of torked a bit in your seat, in the second video, when you asked for a trot, I'm not sure why. But try to post up and forward. Think about moving with him and closing that gap between you and the saddle. Posting should be subtle.
Your horse seems to respond very naturally and easily to your leg cues, which is great. I don't see any resistance there, except for a slight head toss when you ask for canter. Try making a wider circle and look well before you turn (eyes always planning out where you want to go) before cuing the canter, allowing for a smoother transition. It seemed he kind of popped into it, because you asked him a bit suddenly.
Another nice thing is that your horse carries his head long and low, which is actually what you want when you first start a horse English. Since he’s already accepting your contact and flexing at the poll, you can start working him through the back end now and begin achieving a bit more collection.
I’d like to see a little bit more trot out of him. The western jog can be a good starting point for the English sit trot, but at the posting trot, he should be extending a bit more and “tracking up.” This means his hind legs should be moving underneath himself and hitting where the front legs have first hit ground. Ideally you want his hinds to hit exactly where the fores have been. For more trot, push him forward with your legs and open up your seat to allow his back to lift. Some people ride in a "half seat" to create an more extended trot, but that's a little more complicated.
Anyway, once he’s begun tracking up and working through his back end, impulsion happens and you will begin to feel his back lifting and working as you ride. This is what you want. When you see English horses with a pretty “framed” head, this is how they eventually get it (if they’re doing it right). I hope that makes sense.
The "frame" isn't important right now though, just think about working through that back end.
I hope that made sense, if I didn't, let me know and I'll try to explain it better.