It all depends on your definition of “application server”. In the end, I think that splitting hairs over this definition is pointless.
There is a continuum between these two extremes: dumb-client and all the application logic residing on the server and smart-client with little or no logic on server (database only?). Desktop applications fit at one end and web 1.0 sites fit at the other. What AJAX does is shift the focus point on the line between these two extremes closer to somewhere in the middle.
The point is that there are things that the server does well and things that the client does well. Finding that ideal balance is what AJAX enables.