The rules and laws aren't necessarily consistent from state to state. We know this. But neither are the ways we handle emergency health care, including mental health crisis.
Go listen to Episode 136 of Getting Off, for a discussion of how even the simplest distinctions can get real complicated, real quick, when you throw two lawyers and a psychologist together. Then come back here to listen to a few examples of the kinds of crisis evaluations that might happen in a jail.