Looks like the terminology could be locally and is not the official name. I will give it a try:
Shell Press : A Press to used to punch out round shapes for later use in draw dies?
Multi die : A die with multiple inserts or may be a Progressive Die?
Multi die tooling: The inserts going into a multi die to produce different forms of a similar part?
Die Shoe : a Ring or "Shoe" from cold rolled Steel surrounding a hardened die section or sometimes holding together a cracked die for extra stength?
Conversion Press System : ???? Could be just about anything.
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Shell Press, in the old days it was a large custom press used to make artillery shells. But I think you are asking about Clam Shell Presses. They are small table-top units, used for bagging and packaging products.
Die Set = Die Shoe (usually an upper and a lower make a set)