I have thought about this thread a lot as I am one who also learned to ride as a Granny. My impulse was to write that I havent ever done anything that frightened me on a horse and especially never in a lesson - Chose very slow and careful teachers, and saw too many younger beginners who were pushed out of their comfort zone.
So yes an adult student is to some extent in charge of what they do and can guide the teacher and say no to anything that scares them.
However that is not really an honest answer. There is no compulsion to canter - and certainly not until one is well-balanced in the saddle and both you and the RI feel you are ready. Which might include having control of both brakes and steering.
However canter may well make the student less secure in the saddle, so the risk is increased. After I learned to canter, I might still be uncertain whether or not I really wanted to do it - and the girl who took me out in those days said, "Skib, you dont learn to canter by not cantering". This is very true. One can learn things little by little but unless one practises them they will not become easy.
So although the decision to play safe rests with the adult student - so does any decision about learning things that carry a slightly higher risk. I cantered happilly out in the open but for years and years I hated canter in the indoor school. Eventually I booked lessons especially to learn how to canter in the indoor school. And yes I was frightened. But I had paid good money to be taught and when the RI asked me if I wanted to canter, I was resolute and did it.
I learned these skills (against my inclination) because in the UK if one goes to a new RS or goes to try out a horse, a rider needs to be able to ride a hithertoo unknown horse in walk, trot or canter in a school. So that was my objective. I ride out with many young people taking their BHS stages and they are encouraged to get extra tuition for the things they find hard. The RI who taught me to canter in the indoor school went through all the other things that scared me - like trotting over poles - and drilled them till I knew how to do them and they were no longer a worry.
People always say that it may be more relaxed for adults to learn to ride out hacking as I did. But thinking about your question, I realised this morning that out hacking one can at any moment come across something (a barking dog) that may make one nervous and one has to cope with it - handle the horse as best one can and that develops one's riding skills. Thus it is the lessons week after week, year after year that create one's competence. If you give up worrying about speedy progress in your lessons, you will eventually surprise yourself.
Learning how to ride is not a process that can be concluded. Elderly riders may outlive their horses and have to adapt to new ones. I feel much happier now that I have had some fast rides on my new hack and encountered a few excitements. May be I wouldnt have been on those rides and revived my skills if it had not been for an RI with good judgement - an RI who knows the horse and who knows the rider may make wiser decisions than the older student who is wavering.
The deciding factor in all these situations is not the communication between student and RI, it is the ease of communication between the older learner and the horse. It is that which will make you feel safe.