I love lifting, and I agree with this. Even running outside, though repetitive, causes you to constantly scan the ground ahead of you for obstacles (crack in the cement, puddles, dog poop, etc), you have the stimulus of the moving background, cars/people/stray pets to avoid. You may not have to think hard about theses things, but your mind is always processing and responding to the stimulus.

For me, I think I get the most mental benefits out BJJ. I'd wager any martial art with live sparring would give the same results.

When you're doing live sparring, as opposed to just drills, you have to make constant, near instant, adjustments to your defense, look for offensive openings, scan your rolling partner's movements and try to anticipate their next attack, all while being aware of other trainees on the mats (to avoid a random foot or falling body), and working to slow your breathing and maintain composure.

I think what's being challenged here is whether tasks that we consider "cognitively challenging" do actually require more "computing power" than physical exercise does. I guess it's decoupling "cognitively challenging" from "computing power".

Dunno, I feel the effect of lifting is mostly one of persistent clarity.

Whatever wandering thoughts I had bouncing around in my head before a set of heavy deadlifts, all that is gone after, because the effort of pulling up the bar doesn't permit anything but complete focus.

I'd argue against that. If you offer someone a choice between a suduko puzzle or a solidly difficult workout before they're allowed to get one with their evening, I'd guess most people would choose the puzzle. Drumming up the mental fortitude to force yourself to do something difficult is a cognitive challenge.