"I am confused. How much am I supposed to lean forward in the rising trot when riding dressage? Some people say to stay perfectly straight, others say to lean forward. Am I allowed to use my knees as a support or does that disturb the horse or make him tense? What's the correct way of doing the rising trot?"

As in all equestrian disciplines, one of the rider’s principal goals is to stay in balance with their mount, over the horse’s center of gravity.  Since we ride with a more vertical position in dressage (i.e. less angulation of the joints than hunt seat), it’s important to create a position that puts the rider in as much self-carriage as we would ask of our horse at any given stage of  their development.  If you are riding a horse that has just been backed, it may be appropriate to take more support on the stirrup, knee and thigh while remaining lightly seated with the upper torso inclined forward a bit to give the green horse an opportunity to learn to go forward, swing his back, and have only minimal contact with the seat.  However, you would want to straighten your seat to a more upright position as soon as your horse is moving freely forward so as not to propel them even more onto their forehand than they already are inclined to carry themselves as they struggle to learn their balance with a rider aboard. Also, this lighter seat can be used to warm-up a sensitive or hollow-backed horse.

However, I think your question relates more to the correct position for riding a schooled horse in rising trot...  Any tension or gripping on the rider’s part creates tension in the horse. Therefore, it is imperative to ride with as much relaxation as possible while still maintaining the necessary posture most conducive to the proper application of the aids.  The ideal position for a dressage seat is often measured by dropping a vertical line from the rider’s ear, through the shoulder, hip, and heel.  One way I help my students understand how this position feels is to imagine that I could magically remove the horse from underneath their seat.  I then ask them to feel as if  they could land, standing upright, flat-footed, with soft, elastic joints (ankles, knees, hips, elbows) that would absorbing the concussion of landing while maintaining an ideal riding position.  The student begins to understand that they are responsible for their own self carriage on the horse, that they have to be flexible and in balance to avoid jarring themselves as they make contact with the ground as they imagine landing, and that the horse requires the same degree of balance, strength, and flexibility from the rider in order to move freely forward without being jarred by the rider.

Specifically concerning the leg, you would like to feel that your lower body stretches downward from a deep seat, allowing the leg to hang vertically in a relaxed manner.  Ideally, your thighs lie flat against saddle with the kneecap pointing forward.  Your knees should have a light contact with saddle, and never grip, whether posting or sitting.  Also, you don't need to contact the horse with your leg all the way to the ankle as this can result in upward gripping with leg muscles as they tighten to maintain contact.  Rather, the point at which the leg leaves the horse below the calf is determined by the conformation of the horse and rider.  Another image that I find helpful when learning where your leg should fall asks the rider to imagine that their legs are water balloons hanging from the hip, draping along the horse’s sides by virtue of their own gravity and fluidity.  This creates a nice feeling of contact, and downward relaxation through the leg, as well as a sense of flowing with the horse’s movement.  A useful exercise for those who have trouble releasing their pelvis, and therefore the thigh, is to imagine that they are kneeling down, as in church, in a position where their upper torso has to stay upright as they lower themselves to the bench without tipping forward.  This allows the hips to open, the thigh to become more vertical, more parallel to the thigh flap of the saddle, and places the flat, inner part of their thigh on the saddle rather than the larger rear portion of their thigh muscle.  A rider can assist finding the flat part of their thigh by rolling their thigh under from the rear while placing the knee as forward as possible without gripping. When posting the trot, the knee will slide slightly downward along the saddle flap as the rider rises, giving a feeling of sinking down around the horse. 

The correct position for the lower leg is at or slightly behind the girth because it is where the rider has the best balance and has the best effect of the aids.  The horse is more sensitive at this spot rather than further back because of increased nerve endings in this area.  When the length of stirrup is correct, the rider only needs to lift their toes to find the stirrup and the correct position for their foot, placing the ball of their foot evenly across the stirrup bar. The ankles should remain supple and relaxed.   If the stirrup leather is too short, the rider will put their knee up and develop a chair seat, and if it is too long, their toes will point down and develop a fork seat, neither of which will keep them in relaxed balance over their horse’s center of balance. The stirrup leather should hang vertically, and if it is either forward of or behind the vertical, the rider has tension somewhere in their leg that is taking the leather out of the vertical position.

The trot two beat gait with rhythmical, forward, and regular steps taken on alternate diagonal legs with a moment of suspension between the grounding of each diagonal pair.  Rising to the trot encourages the horse to free their back, particularly in the moment of suspension.  The rider sits as the outside fore and inside hind leg strike the ground and rises as they move forward and are in the air. I ask students to bring their hips forward to their hands (without exaggerating the height of their posting) rather than taking their hands back to their hips.  This assists the rider in keeping their balance upright, pelvis and thighs correctly placed, and over their leg, thus discouraging them from using their reins for balance.   Another useful exercise to help the horse and rider gain a sense of symmetry and independence  in posting trot is changing diagonals every so many strides(5 works well) without changing rein, thereby helping the horse’s hind legs step evenly underneath him, and giving the  rider  a sense of balance and proper alignment. Change posting diagonals until the stirrups feel weighted evenly no matter which diagonal you are rising to.  Although this is the long way of answering your question, I think it explains the rationale for remaining in a relaxed, upright position during rising trot.