There is at least one more category, (c) those who are temporarily or permanently homeless by choice. It's not common, but some do prefer the nomadic life and have no desire for a permanent address.

We should take care not to assume that everyone in the "permanently homeless" category is there because of some sort of mental illness or drug addiction, forcing them into unwanted and unnecessary "treatment" until they agree to conform. Not wanting to be housed is not, in itself, a mental illness.

IMHO the only thing that could potentially justify forcing someone to accept treatment (or confinement—their choice) would be if they are manifestly a danger to others. In other words, the current situation without this proposal, except no use of force is justified against those considered a danger only to themselves and not others.