RTC Strengthening
You may want to link to some of the programs you reference that “always work them with the lightest pink dumbbells and in the >15 rep range”. I suspect that the RTC exercises referenced are rehab exercises for RTC injuries. Injured tissue is exercised differently than normal healthy tissue and often specific muscles are worked in isolation, which are two of the reasons lower weights may be used.
Sport Specific
Other, RTC exercises can be specific to sports rather than an injury. They may emphasize eccentric contractions to address the deceleration phase of pitching for instance. They may use PNF patterns to achieve control in functional movement patterns. They work on neuromuscular control and not just muscle strength.
- Tendon vs Muscle
An RTC injury generally involves the tendon(s) rather than the muscle. Because the RTC’s job is to properly set the head of the humerus (the ball) in the glenoid (the socket), and in the proper relationship to the scapula, if the RTC has tears or degeneration and is not working correctly, any attempt to overload the RTC muscles can cause additional damage to the weakened tissue rather than strengthen the muscles.
- Shoulder and Scapular Positioning
The posture of the shoulder/scapula is important when doing exercises to strengthen the RTC. If the shoulder is in the rounded position (from tight pecs or weak scapular muscles) it can put more strain on the RTC tendons, especially when raising the arms overhead. The cuff can become impinged between the acromion and the head of the humerus.
Since it sounds like you are presently doing rehab for RTC inflammation, your therapist can direct you as to the proper shoulder/scapula positioning, the range and the level of resistance that is suitable for your condition.
- Posture
You can check this q/a for general postural exercises and ask your therapist which, if any, would be appropriate to help you improve your overall posture and reduce stress on the shoulder and RTC. Strengthening the lower traps and rhomboids, for instance, are important to good posture, but also in stabilizing the shoulder/scapula so that the RTC muscles have a stable base from which to work.
Once your injury is fully rehabilitated with free weights, resistance bands or cables, and the RTC and scapular muscles are able to stabilize your shoulder joint(s), then general strengthening exercises like farmers walk or deadlifts should recruit more muscles and further strengthen your shoulder/scapular muscles, as long as you increase your weights gradually. And you can continue specific isolated RTC muscle strengthening exercises, increasing the weight as your shoulder tolerates.