I think you flirt only when: (a) you behave in ways intended to intimate the possibility of sex or romance and (b) you intend to make that intention manifest to the other.
This is just obviously wrong in a number of ways. As I've noted before, people often flirt where neither sex nor romance are in serious view, when there is no intimation whatsoever of the possibility of sex or romance. Taylor is persuaded by this account, I think, because he focuses so greatly on flirting as a sexual act; his entire discussion focuses on sexual arousal, and only throws in even romance at the last minute. But does flirting really have much to do with sexual arousal? Flirting is not seduction; you can flirt unseductively (e.g., just in the spirit of fun), and you can seduce without a single moment of flirtation. There is nothing sexually charged about flirting itself, although one might use flirting as a part of a larger, sexually charged interaction. If I step on the elevator with a lovely young lady and tell her that her outfit brings out the color of her eyes, I might well be a bit taken aback if she were to become sexually aroused by that, or if she took this as my presenting myself as available to her for undefined purposes of sex and romance. And we know that this is a possible scenario because you can pick out ten women almost at random and nine of the ten will be able to give you a personal account of how they were taken aback by men doing something along these lines to them in response to what was only a slight flirtation. To intimate the possibility of sex, you don't flirt, you use innuendo; because there are many cases in which they go closely together, some people admittedly have difficulty distinguishing the two. This is sometimes to the great discomfort of those who don't have such a difficulty.
Carrie Jenkins continues the discussion in good analytic tradition by using absurd science fiction scenarios. She also characterizes the link between flirting and sex too strongly. Flirting is not about raising sex or romance to salience, although it can be a part (an incomplete part, it must be said) of doing so, for the straightforward reason that we like eligible partners for sex and/or romance to flirt with us. Jenkins, I think, unintentionally gets closer to the truth when she expresses it once as 'doing certain things I know will make him think about how cute I am'. This is surely a more plausible characterization, precisely because flirting does have something to do with what we call cuteness, both with recognizing the cuteness of another and with showing how cute you are yourself. None of this, of course, requires intimating the possibility of sex, or raising it to salience, although it does raise the question of how closely flirting is tied to cuteness and what cuteness is supposed to be given that it is in itself eligibility for neither sex nor romance.