打采 -- 意思是指狎客给妓女的缠头, (it refers to the headdress / headwrap, (缠头), of prostitutes in ancient China)
Please Note -- my interpretation #1:-
Prostitutes, or women in general, ancient or modern, take great care / attention with their hair which ensures good looks or at least presentability. Something of importance and a daily necessity, especially for prostitutes whose good looks were and is her best selling point.
Therefore, 无精打采, would be to refer to someone, say a prostitute, who is so listless and lacking in energy or good mood that she even neglects to do something so important as putting on a nice headdress / make-up.
打采 could also, following on from #1, refers to an action of a customer of a prostitute. In the old days, rich customers would gift expensive jewelry for the headdress of their favorite "lady"
So, 打采 would refer to the act of the customer putting, (打), the piece of jewelry on the lady's headdress. If the customer becomes so, (无精打采), i.e. listless, and lacking in "mood" or energy that he did not even want to do something so enjoyable as decorating his lady's headdress.