Unfortunately nobody has done tests with different moisture levels to see how it effects the outcome, at least not that I've seen. Something else that might make moist bud press better is that if the vegetable matter, meaning everything aside from the trichs, is moist the resin can't get squeezed in there because oil can't mix with water, at least not well. So the resin may flow over the vegetation and out to the parchment instead of absorbing into it. It's like how you can extract fresh wet bud with a non polar solvent and not get chlorophyll.
I used to do that using toluene. I wouldn't recommend it btw because it's the smelliest solvent around, but I would put some in with wet bud in a glass measuring cup and use a cleaned glass bottle as a plunger to squeeze the solvent around through the bud to get the resin. The bud would flatten right down from the plunging but it got all the resin that way. Some juice would actually get squeezed out and would sink to the bottom and I would remove the upper solvent layer with a glass turkey baster, which are hard to find now but I got lucky at the time. Then I would add clean water to the collected solvent and shwish it around to "wash" the water solubles out, then collect it again and dry it. Worked perfect, amber resin. You have to wash your extracts. People don't do that with butane.
I don't coddle weed when I'm extracting it, I press the hell out of it to squeeze all the resin out. It takes very little solvent when you're pressing weed down into the bottom of a glass, because it gets much more compact. I gave it 3 leachings to get it all.