Should I store a multi-pump pumped or empty?

Store it with one pump of air in the reservoir, no more.

That single pump keeps the seals seated and from drying out, but does not stress them. Never store a multi-pump fully pumped; that is a recipe for stretched seals and slow leaks. Never store it completely empty for long periods either; the seals can shrink. One pump is the universal recommendation in the Benjamin and Sheridan manuals.