The comprehensive answer to your question can be found in several excellent volumes about green-wood working, so we're not going to get a comprehensive answer here. However, in essence both approaches have their merits. Cutting away the pith can reduce (but not remove) the risk of splitting, although some woods, which are less prone to splitting, can have the pith left in. My personal experience is that my best pith-bowls have come from the centre of a length of dried log, where the ends have developed minor cracks which have not traveled too far down the log.
If you're preparing wood for future use try both approaches. Are you able to safely chain saw a log along its length, through the pith? If so, then do this and keep the split log in convenient lengths, sealing the ends.....avoid cutting into small pieces at this stage. If you leave one smaller log intact and seal both ends well, you might be lucky to get a usable pith-bowl blank out of the middle of it.
Les