Well the dress is very non helpful for a critique not to mention it could spook the horse and doesn't provide the necessary grip or action in the saddle. This leads to an ineffective seat and a greater possibility of mishaps- tell your mother if she cares at all for you and your safety that you will only ride in proper riding attire.
Anyways as far as engagement goes - you are a long way (in training) away from being able to really think about increasing engagement in the horse. If you are having trouble with speed, you have not yet conquered even the first step in the training scale, rhythm. The horse should first and foremost be able to go in a balanced way, with a metronomic rhythm. The tempo should be adjustable, as should the lateral flexion and the horse should be relaxed (this is the second step - relaxation or suppleness). The horse should the become accepting of the bridle and stretch to the bit, creating contact without a backwards pull from the rider (this is the third step, contact). These three steps go very much hand in had and all together develop our impulsion or schwung which is related to engagement, "head set" and the over all balance of the horse beginning to shift to the hind quarters. Think of it as the feeling of a "bridge" of energy connecting the horse from his hind legs to the contact through his topline. The training scale continues on to include Straightness and Collection, which are not very relevant to developing engagement as they both require it.
Finally, cheekpieces are like $5. If your tack is ill fitted - don't expect anything from the horse. The bit is resting much too far down in the mouth - so the horse must hold it up, meaning it cannot relax the mouth and jowel and softly chew the bit. I am also not a fan of the baucher bit, a loose ring, dee ring or full cheek are preferable.
Otherwise your leg and posture appear to be good - good luck!