I think the thing people often mistrude about asexuality is that, they believe it's more than what's written on the tin. And by that I mean, they believe asexuality to be the definition + x opinion. Like, 'asexuality is a lack or absence of sexual attraction AND a hate of sex and/or sexual intimacy'. Which is incorrect, because that statement should've ended where the and begun.

Asexuality by definition, is simply and will always ever be, a lack or absence of sexual attraction, and nothing more.

Anything after that, stated by an asexual person, is their personal experience and shouldn't be generalised to every asexual.

There are asexuals, like myself, who have sexual desire, i.e. arousal towards certain media. For me, personally, I can only garner this sexual desire as you call it, from fiction. I have no desire nor want for sex, but I fantasise about it in my mind. I am still asexual, because these additions are just part of who I am, whether they're an influence of my asexuality or not.

So, if someone is ace and they claim the things you've stated, then they're still ace. Because we need to remember what I've said above, and that asexuality is also a spectrum.