I think the first explanation is incoherent - really, what does it mean: '... jumping back to the past... ...where there is no time travelling' ???
I think the explanation has to be that one can only change history through time travel if the altered history is internally self-consistent, from a causal perspective; otherwise the altered history 'collapses' and the original history reasserts itself. In this case, by travelling back in time, the traveller created an alternate history in which he existed as a time-traveler; therefore to maintain causal consistency, his alternate-history non-travelling self (who here was just born) MUST travel back in time to stabilize the alternate history.
But as Bender explains, by winning the election, he prevents the catastrophe that motivated the search for the time code and the resulting time-travel, so he failed to stabilize the alternate history. The original history - where Nixon wins unopposed - reasserts itself. We even see the Planet Express crew - those most closely tied to the time-traveller - start to forget the traveller as their original history memories reassert themselves (though the fact that this does not happen immediately is interesting).
Why didn't the traveller search for the time code while President, you ask (I know I just did)? It is highly likely he simply wasn't aware that it was necessary. So on the date when he was supposed to go back in time but didn't, from that moment the alternate history began to unwind backwards. The unwinding just happened to hit the traveller just after the election when he was on the podium, which is why he started to dissolve just then.
On the other hand, we must also remember: Time-paradox duplicates are always doomed!