you are willing to take a smaller chance of correctly answer "True" instead of a larger chance of correctly answering "False"

Sorry, that's what I meant by "I want to answer True even if it's not correct" :)

Where in my modification, you want the correct answer to be "True", but if this actually influences whether you answer "True" or "False", you did something wrong.

Yours strikes me as teaching the skill "disentangle mixed incentives" - there's an incentive to be correct, and an incentive for the correct answer to be "TRUE". There's also the skill of recognizing which of these two incentives you have control over (the former only). While these are certainly USEFUL skills, I question whether this really helps people avoid Motivated Cognition. On an abstract level, it seems like they might be related, but I don't think it's the sort of exercise that would build an intuitive sense of "Oh, wait, I'm doing motivated cognition here!"

I feel the original exercise, however, does a good job, because it puts the person in a position where they can SEE that they're doing motivated cognition. It lets them get a good look at what it FEELS like to be doing motivated cognition, what it looks like. Once they can recognize it, it's a LOT easier to fight it :)