The Theory of Relevancy
I think there is one important difference between the "relevancy" statements made by Clinton and Bush (as described here by The Carpetbagger Report).
Clinton: “The president is still relevant here ...”
Bush: “I want to make sure — and that’s why, when I tell you I’m going to sprint to the finish, and finish this job strong, that’s one way to ensure that I am relevant; that’s one way to sure that I am in the process.”
The difference: Clinton referred to "The President" in the third person while Bush did in the first person.
In other words, for Bush, it's about his relevance while for Clinton it was about the relevancy of the Presidency as an institution.
(Caveat: Bush's statement was preceded by the following quote, “Sometimes the legislative branch wants to go on without the President, pass pieces of legislation, and the President then can use the veto to make sure he’s a part of the process. And that’s — as you know, I fully intend to do.” So Bush did start out talking about Presidential relevance in the third person, but it eventually came back to talking about himself. I don't have the full text of Clinton's comments to refer to so maybe he also used the first person at one point.)