First a warm welcome. I too learned to ride aged 62. I am still riding 15 years later.
My first all important message to you is that when one begins in old age there is no real obstacle. Providing one has patience and a teacher skilled in teaching older adults.
One conventional rule used to be that you rode with your heels pressed down to prevent the foot sliding into the stirrup, but, believe me, I am going to tell you to ignore this.
We are never going to look like a textbook picture of a rider in the ideal position. The priority for older riders is to learn to balance well on the horse for safety's sake (so we dont topple off to left or right nor tip forward nor back).
The weight needs to be in your seat - you should be able to feel the movement of the horse's hind legs through your seat - One side of your seat will dip and then the other. Your legs should be relaxed down the sides of the horse. Because the body of the horse needs to swing from side to side to make room as its hind legs reach forward. And your feet need to be relaxed as well. One way of relaxing your feet is to spread out your toes inside your boots.
If you sit on the horse with your legs relaxed, and think what it feels like, it will be easier to keep your knees and feet soft when you rise in the trot. Spread your toes out and relax your ankles when rising too.
If you press down hard on your stirrups as you rise, and let your feet move forward, it makes it more likely that your foot will slip into the stirrup. And if you are putting lots of weight on the stirrup you are probably standing up too far in your rise. Our RI told us to roll our hips forward between our hands. Imagine rolling a ball, just a small gentle movement is enough.
When you start to canter you may well find the same thing happens - your feet slide into the stirrups. Again that may be because you stiffen your legs or knees. My own unconventional solution is not to worry too much about it but to ride in safety stirrups. Do you use safety stirrups in your lesson?
If you do, I do feel not worrying is the best approach. My OH also learned to ride in old age and it was 10 months before he learned rising trot - tho he could already trot and canter well sitting. The time scale seems massively important when one is in the first year of lessons, but dont get too worried about this one small detail.
There is one further question. Are both your feet slipping into the stirrups or just one? If it is only one, then you may be sitting crooked on the horse. This is something old people tend to do if given a horse that is too wide for them. When you first get on, look down at your knees - is one further forward than the other? Or is the RI asking you to shorten the stirrups in order to make it possible for you to sit straight across the horse or pony. These are quite complicated adjustments which arent necessarilly helpful to elderly beginners.
But having to have short stirrups may make the riding trot harder for you. I was really able to progress only when a senior BHS judge was visiting the yard and instructed my young teacher to let down my stirrups, and teach me on a horse rather than on a pony. Horse, and saddle and stirrups all need to be right. But on the other hand, your stirrups may be too long -so you are reaching for them with your toes and that is why your feet are sliding down in the stirrups.
So although your question seems simple, it cant necessarilly be answered on an internet forum. But I do welcome you here and wish you well.