This 2010 study might shed a bit of light on your question. It's looking at damage (and then the re-strengthening) of tendons.
Microtrauma can occur when the patellar tendon is subjected to extreme
forces such as rapid acceleration -deceleration, jumping, and landing.
The posterior proximal patellar tendon is subjected to greater tensile
tendinous forces as compared to the anterior region, especially with
jumping activities and deep squat exercises, with forces up to 17
times body weight being placed on the patellar tendon in Olympic
Summarized to me this states (simply) adaption-level exercises produce micro-trauma in the connective tissues (tendons, in this case). Continuing on:
Loading a tendon in a controlled environment free from overuse with
progressive stress improves tendon function.
So the study basically says that progressive weight training helps your connective tissues, which is sort of obvious since connective tissue is what allows muscle strength to actually act upon anything (bone).
Specifically answering your questions:
Is it possible to quantify my joint strength progress?
I'm not aware of any particular way to say that a tendon is x stronger than it was yesterday. Also, "joints" are made up of a lot of tissues. Tendons, ligaments, bursa sacks, cartilage, mini-muscles, etc. The only way I I know of to say my shoulder "joint" is stronger would be by doing an overhead press heavier than I could before, or more handstand pushups, or something to that effect.
Which type of training promotes stronger + healthier joints? Can both things be achieved with the same training style?
Based on the study referenced and everything anecdotal and experimental I know of, it's about progressive programming. You need to increase stress on the joints just like you would muscles, in order for adaptation to occur.
Regarding the safety aspect, most of that boils down to picking the right exercises and performing them properly: the squat, the deadlift, the row (or clean), and the overhead press. Toss in pullups, dips, and bench pressing for a fairly complete system. You could do nothing but those and achieve phenomenal results.
It's a common answer to many questions on here, but I'd point you towards a progressive strength training program which has thought-out increases in volume and weight.
One thing I would very much avoid are isolation exercises. Bicep curls, tricep kickbacks, wrist curls, etc. Unless you have an extremely good reason (backed by a smart person's opinion) stick with compound lifts. Isolation exercises are a terrific way to ruin joints in the short and long term.