i think you already know the answer here, nomos. you've pretty much outlined it already. the terms, in a general sense are rarely not one and the same. there's usually very little difference in what people mean when they say rap or hip-hop and most attempts to delineate the two are wrong in the strictest sense. not always, though. sometimes this is a very deliberate strategy.
for example, purists will often say that hip-hop is the whole culture that the four elements produce in combination and that rapping is only one of those things. they'll also refer to stuff like d4l, hurricane chris, soulja boy etc as rap because "it has nothing to do with hip-hop"; the word rap in this sense functioning as a kind of distancing, derisive term (still, the fact that these artists have little to do with real rapping, either, is a stickier point..)
interesting stuff about the number of elements, too. actually the most interesting thing about the initial post. it's true, they have shifted and changed a lot. for example, production is definitely one these days, extending far beyond the role of dj, which it has to all intents and purposes superseded, and in a completely different way aesthetically, practically, functionally etc. after all, scratch djing is little more than an anachronistic spectator sport that bears little relation to contemporary hip-hop culture now...
i guess the only way you can subdivide rap and hip-hop in any meaningful way, not a cluless one, is by saying that a record can be hip-hop without being rap. take something like the, errr... i dunno... say endtroducing, just for the sake of argument. it's definitely *hip-hop* but not *rap*, same for any hip-hop instrumental album or, at least as far as i'm concerned, any contemporary R&B record - production and aesthetic absolutely hip-hop, very little or even zero actual rapping. conversely, something like grime (mixed biscuits beat me to the punch on this point, but my post is longer and thus better, so i'm leaving it in!), baile funk, kuduro, whatever the hell diplo's getting sweaty about this week generally features rapping, but is not necessarily hip-hop.