Would an accurate summary of this be "humans have a generic, intuitive, System 1 Truth-detector that does not distinguish between reality-correspondence, agreeability, tribal signaling, etc, but just assigns +1 Abstract Truth Weight to all of them; distinguishing between the different things that trip this detector is a System 2 operation"?

That's not how System 1 works in my experience. System 1 is only concerned with modeling of the world and making predictions, particularly of the results of various actions one might make. Its model however tends to be extremely primitive. Also System 2 doesn't have direct access to the model, only the predictions. Furthermore, as far as System 1 is concerned making statements, or even having System 2 believe something, are actions whose consequences are to be predicted.