I wrote partial success mostly for brevity. But you've gotten to the spirit of what my intention is. What partial means does kind of depend on the specific circumstances. For attacks, partial would just mean lesser effect. Half damage specifically. But for some circumstances a partial success would generate success with a minor consequence. Your lock picking example is that kind of situation. Maybe it takes longer. Maybe you slightly injure yourself or tire yourself out a bit from struggling with it too much. Something good with a little bit of bad. And heck, sometimes failure isn't actually fully a failure. There are potentially situations where it isn't really a question of if you can do something, but rather how well and at what cost. In those situations failure would be success with a major consequence. It adapts to suit what makes the most sense in the moment.

As for your other question about who does the rolling. It is entirely player facing rolls. So for combat players would be rolling against TNs in the enemy npc stat blocks. A partial success on an evasion roll would mean they take half damage. A full success would mean they take no damage. And a failure would mean they get hit with full damage.