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My understanding was that warm-up was to get the blood flowing the muscles and capillaries at the skin. Stretching was to get the joints moving, and the synovial fluid flowing, before the joints were asked to do hard, as in high repetition or weight bearing, work. It also lets you detect any potential injury sites with stiffness or soreness.
I am not now, nor have I ever been, a high performance athlete. Not to mention that I can't recall ever spending a night in a Holiday Inn Express.
I do, however, exercise regularly and have done various forms of exercise for various purposes over the years. During that time I have developed my own ideas about stretching.
If what you are after is flexibility, stretching as an exercise rather than a warmup is, IMHO, very beneficial - perhaps essential.
If you are going to make very sudden start and stop movements of relatively short duration, stretching will help prevent those nagging muscle pulls. For example, stretching before playing in your softball or flag football pickup games is advisable if you wish to avoid those horrible groin pulls and such that come from sudden, sharp lateral movement and sharp bending and turning. For throwing I personally think it is absolutely essential (as is technique) and I sure wish I'd done it as a younger man - I might still have some bits and pieces of shoulders that didn't constantly complain about the abuse I heaped on them over the years.
Never saw anyone stretching before weight lifting but I have done very little of that.
Never saw any benefit from stretching for things like "jogging" or brisk walking or bicycling or hiking or paddling or such things.
Stretching is critical for retaining range of motion and the sort of flexibility that lets you get work done--bending, twisting and squatting.
If one is a "performance athlete" stretching pre-event, you're probably going to hurt your performance. If you're just some joe that goes to the gym to retain muscle mass and density as you age, then it really doesn't matter much.
However, the simple answer is to do something to warm up--as AM and the OP note--and then get into the meat of the workout, then stretch afterwards when the muscles and connective tissue are good and warm.
William O. B'Livion
Spouse got "frozen shoulder" which comes from not stretching it out, I guess. Remedy was a couple of physical therapist thumbs painfully breaking thru "adhesions" followed by a lot of stretching. I'm in physical therapy following my 5th spine surgery (a fusion, which is working) and most of that PT is stretching. Must be good - and also seems to be working!