As for making a determination...well, that is pretty subjective I'd say.
But--certainly moms make that determination in various ways. If she keeps saying she's tired, hurts too much, can't go on--and it's not 'just transition' and there isn't any progress over some hours, maybe also a reduction of intensity/frequency of contrax. I've never made the determination of exhaustion--the moms do.
However, even if a mom isn't saying she's done, but she's no longer participating in taking care of herself such as refusing food, fluids, peeing, trying positions or methods to get some rest (or better efficiency of labor), then I might expect to see some signs in the baby associated with maternal exhaustion, dehydration, low blood sugar. "Non-reassuring FHT"--not necessarily scary-bad, just possibly less variability, or rising 10bpm or a bit more from usual, even without baby's greater activity level. In this case I might be the one to raise the subject of whether or not staying home is still what she wants. For some, just reflecting what I'm seeing and hearing from her and baby is enough to help her dig deeper and get on with things. For a few it is support for their decision to get medical help....still, I'm not exactly 'determining' anything--it is the mom/partner who do that.
I'd like to know others' thoughts/experiences on this, though.