If I want a 4” Dia. Core, what size Core Bit do I need?

You would need a 4-1/4” Core Bit.  Core Bits are sized based on their Outside Diameter. They are typically made from 1/8” thick wall tubing, thus, the ID is ¼” less. Hence a 4” Core Bit gives you a 3-3/4” Core. Typ. 6”, 8”, etc.