Should we define? Which cognition? Memory dependent or without involving memory? Which memory? Conscious cognition or any cognition? Higher order cognitive actions like telling a story or mathematical proof?
Instead of using one term for everything and try to define it, it is good to specify a phenomena and define that. Definitions help us to keep things specific.
Some times we try to define an all encompassing concept, and then we end up with a lot of confusion, since that is not the purpose of defining.
What I am trying to bring home is: let us not try to define broadly what cognition is? Let us think of a list of different cognitive phenomena and try to define them, so that we can have better experimental indicators for each of them.