The correct answer is, "it depends." The traits that will be under the strongest selective pressure will be those related to survival in the face of the most common causes of death for that species. If food is abundant but so are predators, then clearly predator evasion will be of supreme importance. If food is scarce, then perhaps metabolism-related traits will be under strong selective pressure. But it's important to remember that just because one selective pressure is stronger doesn't mean that the rest disappear. Thus, the individual's survival is dependent upon really a composite fitness that depends on a countless array of selective pressures, some of which may be larger components than others. The competition between individuals of the same species is really a complex interplay of fitness effects, then.
As for the maxim about outrunning a cheetah, what it's illustrating is that the gazelles are not competing with the cheetah for survival. They're competing with each other. If you want, you could expand it to say "...just the other gazelles and the rest of the herbivores in the environment". If a herd of gazelles and a herd of some slower animal are nearby and a cheetah attacks, one of the individuals from the slower species will probably get caught and all the gazelles will get away. The point is that the gazelles and the cheetahs fill very different niches in the environment, so they don't really compete for the same resources (as a matter of fact, one is the primary resource of the other).
When it comes to determining survival, one talks about fitness only within the context of the population (or, if you want, between species directly competing for the same niche). The less fit members of the population do not survive. Thus, the less fit ones that fail to evade the predator are simply doing a worse job evading the predator than the other members of the population. But it's not only the relative fitness that is important. Imagine that a new predator migrates into the environment, for which none of the prey has any evasive adaptation. Even though there may be a relative fitness between them regarding evading this predator, if even the best individual is not good enough to evade it, they would all have an absolute fitness that is too low to ensure survival. Then the saying breaks down, since none of the gazelles can out-run each other with respect to evading this new predator.
Of course, this idiom falls apart when you start talking about asocial species, since there aren't any other individuals around to out-run.