Telling someone that they aren't asexual is generally interpreted in one of two ways. It could be understood as a clarification of the definition, in which case it will be perceived as being fairly benign, or it could be interpreted as an attack on part of someone's identity, attempting to delegitimate it, in which case it will be seen as less-than-benign. The fundamental issue is who has the right to define asexuality. Does each person define it for themself? Does the asexual community decide as a whole? Does each person decide for themself based on the general framework set early in the asexual community's history? Does some person—either inside or outside—take it upon themself to define it for everyone and then feel it is their right to enforce that definition?
Perhaps the broadest and most inclusive definition of asexuality is that an asexual is someone who calls themself asexual. This definition prohibits people from enforcing their own personal definition of asexuality on everyone else, and it seems to be a necessary consequence of granting everyone the right to self-identify as they choose. However, this definition has a serious problem: it makes no sense by itself and is mutually incompatible with any other definition.
Suppose I were to invent a new word (and identity) called K'liida: "A K'liida is a person who calls themselves a K'liida." Am I a K'liida? I have no idea, but no one can tell me if I am or not; I have to decide for myself. But I have no clue what it is or why I might want to consider myself one. "An asexual is someone who calls themself asexual" makes more sense than "A K'liida is a person who calls themself a K'liida" precisely because we have another definition of asexual at work that people have to decide if it fits themselves or not. But "An asexual is someone who calls themself asexual" is fundamentally incompatible with any other definition of the term because it gives each person the right to consider themself asexual or not based on whatever definition they want to use. The asexual community has constructed our own definition and hope that others adopt it, but we can't force them to. Moreover, in saying that anyone who calls themself asexual is asexual, (if it is interpreted as absolute) we are giving up the right to tell anyone who rejects our definition but still identifies as asexual that they are wrong.
Asexuals have created their own definition of asexuality. Since language is not static, we can create new words whenever we want, or we can apply new meanings to old words. There is no authority on what the correct meaning is (the dictionary tends to be recognized as an authority, but dictionaries don't limit the words in a language and they can't prevent meanings from changing. The standard dictionaries definitely don't include our definition of asexual.) Meaning is determined by usage, and words are valid if there is a common understanding of the meaning between both speaker and listener (or reader and writer.) If I make up a new word, the only way it becomes a "real word" in the language is if a whole bunch of other people adopt it as well. Asexuals have created a word and are trying to get their definition to stick. So far, quite a bit of progress has been made, but at any time, anyone is free to use the word in a quite different way from how I use it or how I, or anyone else, thinks it should be used.
Another issue is that asexual identity and asexual orientation do not completely coincide. Many asexuals feel that they have always been asexual (orientation), but that the asexual identity didn't develop until later. Some have coined the term on their own, others haven't—and even those who did make it up on their own have probably come to have a much expanded idea of asexuality once they learned about other asexual people. Moreover, some people identify as asexual but then later change their minds, deciding that they weren't asexual after all—rather they simply hadn't understood their own feelings well enough, or hadn't been willing to acknowledge them. So even though they had identified as asexual, retrospectively, they feel that they had been mistaken in this—they called themselves asexual, but they were wrong. This is similar to how some asexual people used to think that they were gay ('cause if you're not straight, that's what most people think you must be), but later decided that they weren't because they were equally unattached to the same sex as they are to the opposite one. (Of course, it is possible to be gay and asexual, or lesbian and asexual because of non-sexual attraction.) Not everyone who is asexual (orientation) calls themself asexual (identity), and not everyone who calls themself asexual (identity) is asexual (orientation.)
Despite all of these theoretical problems with the definition "an asexual is someone who calls themself asexual," these problems don't appear to be particularly important on a practical level. When people come to some asexual site wondering if they are asexual, they are almost certain to come across the "does not experience sexual attraction" definition, which has a good chance of significantly impacting how they think about asexuality and how they choose to identify. Since this has a strong influence on who does and who doesn't choose to call themself asexual, it effects who is in the asexual community, who is active on the forums, and who is involved in other places where asexuality is discussed. For a lot of people, the "does not experience sexual attraction" is part of the reason they identify as asexual, but an even bigger part is that as they read about the experiences of people identifying as asexual, they find something that fits close to their experiences, something they can identify with after such a long time of being bombarded with messages about sexuality that didn't fit with their own feelings and no messages even recognizing that there are people like themselves. Reading about the lives of others identifying as asexual provides a sense of validation for such people, and this has a strong effect on whether some particular person, after finding AVEN or some other asexual site, chooses to identify as asexual or not. Thus, the definition puts selection pressures on who identifies as asexual, who identifies as asexual puts selection pressures on who is active in the generation of asexual discourse, and this puts selection pressures on who subsequently identifies as asexual. Within this context, there will be a strong connection between people who call themselves asexual and people who experience little or no sexual attraction, while at the same time giving each person the right to self-identify as they choose, hopefully without feeling pressured to go one way or the other.
A practical consequence of the definition "an asexual is someone who calls themself asexual" is that it helps shape the direction of asexual discourse. The intent is to push discussion on the forums in certain directions and away from others. People spending time trying to explore their own thoughts and feelings, thinking about their own experiences, trying to better understand themselves—these are the sorts of things that people think about when trying to decide if they are asexual or trying to figure out what being asexual might mean for their lives. People spending time telling others that they can't be asexual, forcing others into narrowly defined asexual molds, arguing about what real asexuality is—these directions aren't as helpful for real people. They seem more about people trying to use a narrowly defined asexuality to inflate their egos than helping people think about their lives.
I recently got a chance to talk to David Jay about definitions of asexuality and his comments were very helpful. He said that in the asexual community, there are two ways of defining an asexual person. "A person who does not experience sexual attraction" is the definition that is primarily intended for people outside the asexual community as a way to introduce asexuality to people. "A person who calls themself asexual" is the definition used inside the asexual community, and these two definitions are framed the way that they are to steer discussions in certain directions and away from others. As I think about these definitions in this way, they make a lot of sense. From a theoretical perspective, I think I've argued effectively that the definition "an asexual is someone who calls themself asexual" has a lot of problems with it. But if we think about it not in terms of theoretical accuracy but practical utility, it makes a lot more sense.