Even if the first figure is already leaning, and you only move back a couple of inches, it still puts more weight on the hands by moving back. As long as the two lines are the same length (seat to shoulder, shoulder to hands) in both situations. I may be harping on this too much, but if we tell people "move the seat back to take weight of your hands", they do that and realize that it really doesn't, then what?

I agree with you, keep the saddle level or even up can help tremendously compared to tilted down, pushing you against the bars. And also it's a good point that closing the torso-hip angle your core can take up some of the weight. To be sure, moving the seat back does help sometimes - I just rebel against the common reasoning that "it takes weight off the hands", and "it's where you'd be if you balanced without a saddle" because both reasonings are wrong even when the result is correct. It's a matter of overall fit, not weight distribution.

There's another thing that moving the saddle back can do for you. Think of the angle of your leg and the angle of the power stroke on the crank when you adjust it back (and usually lower to compensate). The power stroke begins slightly more counter-clockwise as a result, having the effect of pushing back against the body away from the bars. I think that's helpful. But it's not static as we'd expect with just weight balance considerations, but changing as we become more or less fit and can apply more continuous force to the pedals for example.