A good start to thinking about rest times is taking a look at the context of the movement as well as your goals.
Are you doing a particular movement for strength, for cardiovascular fitness, to build muscle or to drill the movement? Drilling technique is best done with longer rest periods so that your body learns good and clean technique instead of tired and potentially sloppy technique. For strength and building muscle, the body responds to overload. This overload stimulus can come from different sources. Doing more is an overload stimulus, so if resting more allows you to do more reps, then resting more is a good idea (within the limits of sensible training of course, it does not make sense to rest so long (>5mins) that the body gets cold). However, doing more work in a given time or in less is also an overload stimulus. Therefore, you could also track and manipulate your rest times to challenge your body to complete either the same amount of work in less time than before or more work in the same amount of time as before.
What are your overall goals? For the vast majority of beginners and early intermediates, rest times do not need to be an important factor. Simply rest long enough (typically 1-3mins) until you are ready to do the next set with full attention and full power. For athletes who are mid-to-late intermediate or advanced, you might want to start thinking about your weak points. If you have a general problem with cardio or work capacity (doing enough work in a workout, or getting tired in the workout easily), then working to slowly shorten your rest periods over time could be a good tool in patching up this weak spot.
It may also be useful to consider what phase you are in with a certain movement. Let's say you have been able to do muscle ups for months and you have solid technique, if you record a video of a set of 10 muscle ups, rep 10 looks exactly the same as rep 1, just as clean. You then are considered to have stable technique which does not deteriorate as you get tired. In this case, your limiting factors in your rest periods will be overall systemic fatigue (how impacted your body/heart/breathing is as a whole from the previous set) and local muscular fatigue (how tired the associated muscles are). However, let's say you just learned to do muscle ups a week ago. In a set of 5, each one will look different and they will most likely look worse the more tired you are. In this case, you should definitely rest longer in order to increase the number of clean or good reps that you do. The more you master the movement, the more flexible you can be with your rest periods.
Rest times are determined by you in The Movement Athlete App. We are giving you a 'suggested rest time' and this is the time that will be used in calculating total workout time. But you always have the option to rest more or as long as is needed to do the next set, with the disclaimer that long rests will add to total workout time.
Why? Rest can be very individual. Some people are good to go on basically any exercise after 30s. Other people need 3 mins. Age, training experience, other activities, work, time of day, so many things can affect it and at this stage, it would be impossible for the app to account for all factors. But in the future, we are looking into improving and implementing those algorithms.