Binary Neural Network Classification or Multiclass Neural Network Classification?
I think you are making things more confusing then they are. 

Binary

In this case you have two possible outputs: 


Obama = 1. 
Not-Obama (who in this case can only be Romney) = 0. 


Multi-Class

In this case you have k possible outputs, for example when k = 4: 


k = 0: Obama
k = 1: Romney
k = 2: Clinton
k = 3: Bush


There are approaches to tackle multi-class classification as binary classification which are called One-vs-rest classification and One-vs-one classification, other classifiers, such as Random Forests, are able to deal with a multiclass setting in a natural way. For a brief report, see here.