Start with the definition of "customer." That's the person or people who make or strongly influence the buying decision for whatever products and services you sell. If you work through a staff augmentation firm as a contractor, that company is your customer. Its customers are whichever individual or group of individuals within the client firm decided to use it instead of its competitors. They are also your customers by proxy.
The whole company you're serving isn't the customer; it's the wallet -- the provider of money.
From the perspective of working with each individual to assist them in solving their problems, I personally continue to call them end-users or business users. Technically, they should be called "consumers" as they're the ones who consume the services you provide. They play a very limited customer role; if you and your colleagues do a bad job, their complaints could result in a decision to make a change. They're your customers when you're referring to them in that context -- but only in that context. Other than that, end-user is as good a term as I've run across.
This is, by the way, a theoretical analysis. If you try to put it into practice from where you sit, you'll likely put yourself on the receiving end of puzzled looks unless you're in the sort of bull session that gives you the conversational room you'd need to explain things.
Otherwise, I'd say you should adopt whatever vocabulary your customer -- the services company through which you're working -- prefers.
Or if you are selling your services directly, call everyone your customer as a courtesy, making the tacit assumption that the contracting officer does care about the quality of service you're providing to the end-user community.
This story, "Who is help desk supporting: customers, end-users, consumers?," was originally published at InfoWorld.com. Read more of Bob Lewis's Advice Line blog on InfoWorld.com.