Your horse is definitely good looking *want*...
First off, HOW can you ride in shorts?! Whatever you're doing to not get saddle sore, please share!!
I'd love to ride in shorts without getting a nastyyy saddle rash.
Ok, actual critique...
You need to stop kicking. I wouldn't want him to go any faster, since his trot is already fast and choppy anyway!
Instead, whenever you feel him break gait into a walk, squeeze with your calves and heels, and if he still doesn't listen to that, then make the kissy noise, and if he still doesn't listen to that, then you can give him a kick in the ribs
AND NOOOOOOOO YOU DON'T NEED SPURS!! HARSHNESS ISN'T THE ANSWER!! YOU'LL END UP WITH A HORSE THAT'S DEAD TO THE LEG!!! Spurs are for lateral movement, not for more encouragement to go forward. Once you and your horse are sidepassing and doing reiner spins, then you can use spurs.
Work on your seat. Bareback or stirrupless riding helps that a lot!!
Like others said, your canter looks much better than your trot. (your canter looks good!)
Work on your leg placement too. There are places where you are depending too much on your stirrups, and there are also places where you are getting chair seated. Remember: Ears, Hips, and Ankles should all be in a straight, vertical line! And heels down!
Remember quiet hands! Your hands are pretty bouncy. Try to keep the reins in one spot when you're not doing anything with them.
I haven't watched your other videos, but from what I read you're making good improvement!
I think with some more work you guys will make a great team