That is a cute horse and though she wasn't offering you a lot of foward to post off of, she does move nicely through her back and has really nice rythmic trots. I think you guys will be a really good match! I like that little mare. I bet she has a lot more in her, too.
Your position is surprisingly good for so few lessons,too. your lower leg is quiet but is carrying its' fair share of your weight. You sit nicely upright and your hand position isn't bad. I did notice, not surprisingly for a beginner, that you are not able to keep your hands still relative to the horse. So when she changes speed suddenly or bobs her head, and you lose your balance, the horse gets bopped in the mouth. That has to do with you having your elbow 'locked" . It's nice that there is some bend in it (as opposed to many new riders who have their elbows as straight and stiff as rolling pins), but your elbow needs to kind of open and close as you go up and down in your posting so that your hands stay the same height off the horse .
One thing you can do is tie a piece of string to one or both of the D rings which is about a foot long. When you gather up your reins, also hold the little string so that once your hands are positioned, the slack is out of the string . Now, as you post, you dont want the string to go slack and snap tight, but rather to maintain a constant tightness. (keeps your hands in same place.) You can easily let go of the string if you need to.
I think you have a really good start and I wish you continued success.