Had the barber or whoever cut the hair not kept it, it would have ended up as trash anyway. To the best of my knowledge, trash becomes public domain the moment it hits the curb. The moment you discard something, it stops being your personal property. If your worried about others having information about you like your DNA or fingerprints, then you need to take steps to assure you control things with that information on them. For instance, if your someone as important as Neil Armstrong, you could find a barber that would collect your hair trimmings so you can destroy them yourself. For instance taking that bag and burning it.
I can think of no reason why I would WANT to keep my hair after the barber has cut it off, but unless I willingly give it away (for example, to be donated to Locks of Love), it is still my hair, and thus my property. Just because it has been cut off, that does not automatically cause it to not be mine.
If another part of me were cut off, such as a finger (or less gruesome, a fingernail clipping), anyone who passes by cannot simply claim it as theirs, unless I allow them to.
No, my hair, cut by another person and falls to the floor, is not a public item. It is the property of the person/people/company who cut it, because my hair fell in their shop. If, however, my hair were cut in someone's home, it would be the property of the person who owns the home.
No, cut hair is not a public item. When I enter a hair salon, I expect to be treated politely and courteously. I do not expect for my hair clippings to be collected by anyone who saunters in and scoops them up. If a salon allows hair clippings to be collected, customers should be informed of this beforehand.
Once my hair has been cut I may or may not want it anymore, but that does not make it a public item. Buying and selling hair is a valuable business and if I no longer retain ownership, this is considered stealing from me. My hair belongs to me, until I give it up or sell it.
No, my hair is not a public item even if it is cut by another person and it falls to the floor unless I agree to throw it away. It is still my hair to own and control until I decide to throw it away. After I decide to throw it out, it is a public item that can be taken by anyone.