So then would it be correct (or more correct) to say that I could collect my horse by holding my rein position steady, and push my horse into the bit pressure, rather than putting pressure on the reins and keeping leg pressure up?
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Rhythm - riding forward in a good balance
Relaxation - the horse begins to relax into the rhythm
Contact - the horse stretches to the bit and becomes "round"
Schwung - the energy from the hindlegs to the bit becomes a strong connection allowing the horse to really use his back
Straightness - the horse allows you to ride him on your line while bending his body, he becomes more supple and accepts all your aids
Collection - the horse takes aids from the seat (actually it is a lot like how bsms explained with weight shifting and balance) and his balance shifts over his hocks, the steps become more elevated and expressive, and shorter (but the "air time" does not change). With this his body compresses and he is able to coil like a spring of energy, making him quicker, more agile and more balanced. It is like running with your belly out versus running with your core engaged.
The whole shebang takes a year or so before you're kind of working on collection and after that the next 15+ years of the horse's training are focused on refining and refining the collection for ever and ever. At least in dressage.