Each of these methods uses a different “attack” on the original toner bond and gives each technique a unique aesthetic.
The solvent method dissolves the polymers or waxes to release the carbon. This leaves no binding agent so the carbon particles must be embedded in the new transfer surface. This dictates that the solvent transfer method be done with a porous receiving surface. Toothier papers and wood, work well for this method. Its main drawback is the danger involved in using the powerful solvents required (either acetone or xylene).
The thermal/pressure transfer basically reintroduces the original binding environment of the fuser in the printer or copier. It heats the binders to the point where they are fluid once again and ready to bind to a new surface. It requires that the receiving surface be hotter and stronger than the paper so that the binder/pigment can flow onto it and then the paper is ripped away. This dictates that the receiving surface be something like metal or glass, which is probably one reason why this is an uncommon method.
Acrylic gel transfers basically use acrylic binder without pigment as glue that pulls the carbon and binder to the new surface through a stronger bond than with the original paper. The polymer chains in the acrylic probably chemically interact with the polymers in the toner to interlock. I did not find any scientific studies into this, but it would be interesting to see the chemistry and physics behind this technique.
The acrylic gel medium transfer method is the one I will go into in more depth. It started as a complicated process that took much time. You would basically lay up many coats of acrylic gel medium on top of the toner print/copy to create a thick, rubbery membrane on the paper. After it had dried, you would then bathe the paper and membrane in water and rub the paper off of the membrane, leaving the toner embedded in the membrane. You could then apply the membrane to whatever you intended (be it a collage, piece of paper, or anything else). Not too many people seem to have taken up this technique since it was so time consuming and gave you a thick, rubbery image at the end.
Paul Fujita, in 2003, was the first to document a much more streamlined acrylic gel transfer method. He talks about how he had heard rumors of an easier derivative of this technique, but no one ever taught it and no one ever documented it. Apparently the word got out about his new method and people kept asking about it, so he created the first how-to guide for it. (Fujita, Interview with Paul Fujita on Acrylic Gel Transfer Technique) This refined technique is non-toxic and works on almost any surface, making it the most accessible of the transfer techniques.