What matters, I suggest, for distinguishing flirtations from these other actions, is the fact that in any genuine flirtation there should be an element of playfulness. Flirtation is, of its essence, playful, and intentionally so.
I think this is quite clearly false. Of course, people can flirt playfully, and probably enjoy themselves more when they do, but it doesn't take much to find people who don't. Some people who flirt are simply drunk; and there are all sorts of reasons not to consider drunk people playful. It takes a certain finesse to be playful, a bit of wit, and for most people alcohol is not conducive to either. For some people flirting seems to be more painful than playful; shy people, for instance, are only occasionally playful when they flirt. Jenkins notes this latter point; but apparently thinks that 'playfulness' need not be enjoyable. This might be so, although I think it doubtful, but it's not enough to get around the problem of painfully shy flirtation. Talking about playfulness also overlooks how desperate some flirting is; and desperation and playfulness don't seem to mix well.
Moreover, playfulness doesn't make much sense of most of the things people do and say when they flirt. It's difficult to see what's playful in saying, "Hi; I like that shirt; it brings out the color of your eyes," and that's sometimes all it amounts to. Saying something like this playfully would probably be counterproductive; it would make it sound less serious and straightforward than it was intended to sound.
Most people seem to spend most of their flirtation hoping that the other person will pick up tiny little signals rather than (as Jenkins makes it sound) setting out with great resolution and confidence "to raise the issue of romance/sex to salience" (as Jenkins puts it). Most people are not very good at it, and aren't trying to get much out of it; they're just trying to catch the other person's attention, or have a little fun, or express how gee-golly-gosh the other person is.
It's also probably not true that flirtation, as such, has such a direct relationship with romance and/or sex as Jenkins thinks. For one thing, not all flirtation is for the purposes of romance or sex, or even for making it salient, whatever precisely that might mean. Genuinely playful flirtation, in fact, is often very ambiguous in this way -- some people flirt just for the fun of it, and not because they have any sort of romance or sex in mind. They might even be annoyed if the possibility were raised, because it would break the fun of a harmless flirtation that no one thinks is going anywhere.