"Asexual" has two plausible derivations, which I will label as (1) and (2)
1: Asexual means "not sexual."
2: Asexual, by way of analogy with homo/hetero/bisexual, roughly means "sexually attracted to neither males nor females.
I will set aside issues of gender binaries involved in (2) (as the matter is not relevant here), and I will assume that (2) is understood in relative rather than absolute sense. ("Little or no sexual attraction" rather than "no sexual attraction ever.") I've thought about this matter a fair amount this week, and I've become dissatisfied with my last post. So I'm going to try and start over in attempting to understand these issues.
First, there is the historical question. While I don't have any strong evidence for the following account, I think it generally points in the right direction. So here's the first question: of (1) and (2), which came first? It seems obvious that "not sexual" is, in some sense, prior to the other. A number of people have independently coined the term "asexual" either to describe themselves (or others), and my impression is that they almost always mean it in the sense of "not sexual." They have some sense of seeing sexuality around them, feeling that they don't fit that, and having a sense of being "not sexual." The most obvious term for this seems to be "asexual."
However, there is something of a political problem with this: it's vague, different people are going to have different ideas of what "not sexual" would mean (or not be sure what it means), and so on. As a consequence, another definition seems to be necessary to communicate about asexuality. The definition that has come to prominence is "Asexual: A person who does not experience sexual attraction."
This definition was never intended to be a definition contrary to (1), and a number of people have held both at the same time. The fact that someone can hold both at the same time seems to be demonstrated by the number of people who come to AVEN, see the definition "Asexual: A person who does not experience sexual attraction," but in deciding if they are asexual or not, are still wondering about the "not sexual" issue. For instance, they see AVEN's definition, but wonder, "If I masturbate/get crushes on people/find people aesthetically attractive/watch porn etc., can I still be asexual?" Now, AVENites will tell them that none of those things necessarily make someone sexual. This will be defended in one (or both) of two ways: by appealing to AVEN's definition (none of those things is sexual attraction) or by appealing to an identity model of asexuality (only you can decide if you're asexual or not.)
Note that these questions only make sense if they are thinking of "asexual" as meaning "not sexual." The fact that they have seen AVEN's definition has not changed the fact that they are thinking in terms of "not sexual."
Now, when we talk about people being "not sexual," the question arises, "Not sexual in what sense?" If you try to come up with necessary and sufficient conditions for what it is to be sexual (and thus, to be not sexual), you'll run into disaster and the whole matter tends to not make a lot of sense. However, I don't think that is really what "asexual" is supposed to cover. The sense in which people are "not sexual" is intuitive: some people see sexuality around them, they see it performed and enacted in culture, in relationships, in the media, etc. and they have this sense of not being that. There is this intuitive sense of being "asexual." I think that it is fundamentally this meaning of "not sexual" that is intended in (1).
However, the prominence that AVEN's definition of asexual has come to play in the asexual community has created a situation where the are people who feel unable to relate to some aspects of sexuality, they feel strange because they do not feel certain things that it seems almost everyone else feels, and they feel that "a person who does not experience sexual attraction" makes sense of this experience; they feel that the definition fits, but they don't have an intuitive sense of being "not sexual."
Because there are a number of people in the asexual community like this (myself included), something of a wedge between (1) and (2) can be created, even though it was never intended. This is, I think, a matter worth exploring further.