Characterizing a different definition than yours as a "dual use" implies that one definition (yours) is authoritative and must supersede all others.
The definition of intensity as load (absolute: kg, lb; relative: %1RM) should at least sound familiar. It is the definition of strength & conditioning scholarship, that has existed in the English language for the better part of a century, and that is international in scope.
Your definition, that I am familiar with, is nascent & nebulous, pieced together from some very recent bodybuilding literature (the scholarship of which has existed since the mid-2000s).
I am happy to meet you where you are with your definitions, but do not expect me to validate your ego (outsized in bodybuilding) by adopting your pet definitions of terms like intensity, failure, effort, etc. as if you speak authoritatively on the matter.
Gentlemen, I enjoy this discussion very much, please allow me to respectfully interpose my thoughts here:
Intensity, by the Oxford definition: the measurable amount of a property.
Intensity, as it relates to exercise, following the Oxford definition: the relative degree of momentary effort produced by a subject during a bout of exercise. (my definition, adapted from Arthur Jones')
I will also share this more accurately measurable definition when referring to the intensity of a single bout of exercise (a 'set'):
"Intensity is inroad over time; the rate of fatigue," (Ken Hutchins) where inroad is defined as the momentary fatigue generated during a bout of exercise, realized as diminishing force output. (Ellington Darden, PhD)
I believe accurate language is paramount. One mans subjective intensity, failure, etc could be very different than another. I think you both have acknowledged this as well. Having an accurate, clear definition to which we base further conclusions upon is very important to avoid misrepresentation and discrepancies in scientific understanding.
Otherwise, you get Brad Schoenfeld "training to failure" research. Footage of Schoenfeld training "to failure" is laughable at best, see the late John Meadow's YouTube video with him if you would like a good chuckle. You could not call that high intensity by the definitions I have proposed here.
Need not agree nor adhere to my definitions, just thought I would share my perspective.
As it relates to "RIR," I must share this excerpt from what I deem good science:
"Ability to predict repetitions to MF is not perfectly accurate among most trainees though may improve with experience. Thus, RIR should be used cautiously in prescription of RT. Trainers and trainees should be aware of this as it may have implications for the attainment of training goals, particularly muscular hypertrophy." [1]
I encourage you to try this with your clients/training partners/buddies, ask them to go to 2 RIR and then instruct them to keep going. With some verbal encouragement, I find that under my instruction the trainees almost always squeeze out more than they thought. Like
@Type-IIx implies in his most recent post, a multitude of psychological factors can likely effect the perceived "RIR" with significance.
Additionally, in my opinion, repetition count is arbitrary at best, since unless specified, which it almost never is, there is no standard of cadence to quantify time under load. If one were to prescribe a trainee a common goal rep range of 8-12, this could result in 15 seconds TUL, 30 seconds TUL, 60, 90, 240... Hopefully you can see the problem is quite clear when put like that. Go to any commercial gym and observe the vast variance.
Tried to put a conscious effort in this to not speak with implied authority. You both have lot of quality posts. I'm new to this forum, looking forward to engaging with similarly enthusiastic individuals.
References
[1] Steele J, Endres A, Fisher J, Gentil P, Giessing J. Ability to predict repetitions to momentary failure is not perfectly accurate, though improves with resistance training experience. PeerJ. 2017 Nov 30;5:e4105. doi: 10.7717/peerj.4105. PMID: 29204323; PMCID: PMC5712461.
Further reading, newer work by some of the same researchers:
Armes C, Standish-Hunt H, Androulakis-Korakakis P, Michalopoulos N, Georgieva T, Hammond A, Fisher JP, Gentil P, Giessing J, Steele J. "Just One More Rep!" - Ability to Predict Proximity to Task Failure in Resistance Trained Persons. Front Psychol. 2020 Dec 23;11:565416. doi: 10.3389/fpsyg.2020.565416. PMID: 33424678; PMCID: PMC7785525.