The others you name were not generally known by other names ("Bill" McKinlay, anyone?), and in any case James and William have always worked as type-ins for Messrs Carter and Clinton in this quiz. The other exception is "Jack" Kennedy, for whom that name is also a type-in for that answer.
You don't use diminutives for Taft, McKinley, Pierce, Tyler or Kennedy so why do so for Carter and Clinton?.