Alternatively, they are using a vague term in a slightly different way than you are.
Asexual can reasonably be used to mean people who are indifferent to sex, people who find sex actively unpleasant, people who are not attracted sexually to anyone in particular, people who don't experience arousal, and other related groups.
It seems silly to include people who are in fact sexually attracted to other humans, but simply choose not to pursue it for one reason or another. It just makes the term far too vague to be useful. Yeah, I can understand humouring people like that who claim to be asexual, if only to avoid drama. But when discussing actual sexualities, something that's an innate condition rather than a choice, such a definition would render it meaningless.
Some definitions of asexuality refer to a sexual orientation analogous to hetero- or bisexuality. Others speak to other aspects of one's attitude/preferences towards sex. If you are in a context were precision matters you can always specify which one you mean, the same way we do with every other word with multiple related meanings when we need to use it technically.
Anyone could rightfully claim asexuality under that definition simply by choosing not to pork anyone.
Sort of? I think if you really are sexually attracted to people, and enjoy sex, then choosing to abstain is not exactly asexuality. I can't really read your mind though, so if you lied and told me you were not attracted to anyone I wouldn't really know the difference.
But either way, I don't see why people "claiming" asexuality is a problem*. It makes the word less exactly rigorous, maybe, but most of the vocabulary regarding sex and sexual orientation is non-technical and has vague definitions and complicated edge cases.
*I can imagine someone who is asexual bothered by someone else claiming the term, but I don't think that's what you have in mind.
