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For heavier strength work - rest as fully as is possible / needed for you. Generally 3+ minutes minimum. I'm usually 3-5, 5+ for when things get heavy, or I'm on the last few of a large amount of sets.
For hypertrophy + assistance work - generally shorter rest periods, as you say, more stress, more hypertrophy.
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when things get heavy
I can't speak to the medical/anatomical side of things, but surely there is a very useful psychological benefit to experiencing yourself pushing your treshold up another notch. A boost of confidence that will hopefully carry through to the hypertrophy work you will do at the end of the session, so a net benefit overall even if you were indulging with 3 - 5 min rests during the T1 exercise at the beginning of the session.
With that in mind is It better to rest fully then go into the next set fresher or stress the muscle more by jumping to the next set quickly.
I was under the impression stressing a muscle is what made the tears to increase muscle size and strength.