Client to me has an element of "ongoing relationship with the seller" in its meaning: if you go to the store to buy a box of matches, you're just a customer, but if you always go to that store because you know you'll get good service and good prices, you're a client. (However, the dictionary does list "customer" as one of the meanings of "client", so they are very close in meaning if not identical.)
I would say that someone who pays a monthly fee in order to have continued access to an online service could be called a client.
Some suggestions for your terms:
malicious bot = Abuser
entity using programming tools = accessor
non-using public = public
registered but not paying = user
payer = client