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.
would be quite a good start for understanding sexual innuendo, if we place the emphasis on the intimation of possibility that Taylor does; but it misses flirting as such entirely -- and I'm sticking by the 'obviously', since I think that when we consider the broad range of cases in which people flirt that this pretty clearly doesn't give us any insight into flirting. On the issue of playfulness, I've also gone on record denying that flirting is intrinsically playful -- although, possibly, Taylor's phrase "an air of intrinsic playfulness" might be felicitous, since something can have an air of intrinsic playfulness without being intrinsically playful, 'an air of' usually indicating an appearance. Even then, however, I think to put much emphasis on it as a clue to the nature of flirting would be to confuse a mark of success at flirting (flirting well) with what you are actually doing qua flirting (flirting as such); one can say interesting things about flirting from this angle, but the real character of flirting and flirtatiousness has to be discovered elsewhere, just as an account of how to live well can be a little fuzzy about what it is to live.
In any case, Kenny Pearce made an interesting comment:
I think the confusion may be caused by the fact that there is a subgroup of flirtatious behavior that might be described as "seduction in play" or some such. That is, seduction involves a serious (and manipulative) intention toward sex. As the degree of seriousness is lessened and replaced by playfulness so that we reach a range between merely teasing with no intention toward sex at all and playful, rather than serious, intentions toward sex it becomes an uncontroversial instance of flirtation. I think, however, that the confusion enters in the belief that this is a complete account of flirtation, when in fact that word has a much broader semantic range. The innocent end of the spectrum I've just described is a playful imitation of seduction which is (hopefully understand by the other to be) lacking the crucial element of seduction, namely a serious intention toward sex. There are, however, instances of flirtation that are far more innocent, far more subtle, and far further removed from sex, than this.
I think it's quite right to think that flirting has a broad semantic range. I don't think flirting is opposed to seduction; there will be cases where flirting is instrumental to a larger seductive project, and where the intentions involved in flirtation are quite serious, and recognized to be so. There will also be cases such as Kenny suggests here -- 'seduction in play' is a good name for them. There will be yet other cases, more Lady-Susan-like, where narcissism has various roles to play (from being the cause of the flirter's 'flirting with' the idea of sexual or romantic attachment to flirtation's being a purely instrumental behavior to put the narcissist in the spotlight again). There will be yet other cases where desperate for positive attention will be significant. Flirtatiousness may be occurrent or habitual, serious or playful in intent, pleasurable or painful, spontaneous or cultivated, done well or done badly, for sex or for romance or for the mere fun of it. (Jenkins wants to distinguish flirtatious behavior from flirting; I don't see the value of the distinction, but those who favor it could reformulate the point in terms that take the distinction into account.) Flirting is a behavior that can be interpreted differently in different contexts even when exactly the same things are done. For instance, the most common contexts for it certainly are cases where the possibility of sex or romance is in some way intimated; but there are other contexts as well. This is one contrast with something relatively one-dimensional, like sexual innuendo or romantic compliments. Jenkins is quite right in her original paper to distinguish flirting from things like these; one can do them without flirting, and one can flirt without doing them. It also distinguishes it from a multi-dimensional aspect of human life, like seductive behavior, which despite its multi-dimensional character is narrow in purpose.
(My own inclination is to treat flirting in terms of a theory of taste, in the eighteenth-century sense of the phrase; but that is hardly surprising, since I'm inclined to treat lots of things in terms of a theory of taste.)