In my own research, the main focus is to take a (usually known) property and to rephrase it so that the new statement is self-dual. The analogy would be that you "understand a pattern only if you know how to phrase it in a self-dual context".
None of these different approaches necessarily try to presuppose or subsume each other, but they hint towards your outlook and interests. That is why independent discoveries are usually "clearly" independent, in the sense that the whole framework is usually different.
This happened for example with Lawvere and the eventual standard definition of a topos; so too with Grothendieck toposes and the eventual standard definition of a topos. Newton and Leibniz would be the more accessible example.