An ASIC is a tailor-made suit, normal chips are ready-to-wear.
Being a CPU doesn't disqualify a chip from being an ASIC. Being a general-purpose one designed for a wide market does though. Every chip has an application, that's not the point. The point is whether the designer has a narrow market. In the days of massive toolings costs, this was the big differentiator.
Once tooling costs reduced, you can make a chip for your project -- your application. A CPU for a particular job -- say an experimental LISP machine or a Connection Machine -- could easily be an ASIC.
Perhaps the best way to think of an application-specific chip is that it satisfies both of these:
- You can't buy a suitable replacement from the ready-made chip selection
- You can't sell any extra you might make because nobody else would want them
As a term of art, it's not black and white and categorical.
So "ASIC" is neither a "physical" nor a "functional" concept: it's a concept around who specifies it. If the buyer is specifying, it's bascially a custom-chip, an ASIC. If the seller specifies, they are hoping they'll sell lots to lots of different customers.
(If you have an enormous internal market (Ford, Boeing, Google), are your internal "ASICs" actually general-purpose? Each organisation will have its own terminology for such a case. It's not the general situation.)
We know what a tailor-made suit is; we know what a custom machine tool is. An ASIC is a custom chip for a project. Just think of the conversation:
Manager: Can we get a part which does everything we want and fit on the board?
Engineer 1: Texas Instruments makes one but it does a lot of other things too and the package is too big.
Engineer 2: Intel makes a chip which does most of it, but we'd need to add some other parts to cover all the functionality.
Engineer 3: I think we should get an ASIC, it will do just the right things in the right package.
Manager: Okay, but I'm worried about minimum production run and cost.