Can I drive home a vehicle whose listing specifically uses the term "clean designated" from its storage location?

"Clean" in the title-status field means the same thing as clear or clean title elsewhere — the vehicle was never branded salvage. Combined with a verified run-and-drive status, you should be able to drive it away after completing your purchase, subject to the storage facility's own pickup and release procedures (which can include scheduling a release window or showing ID and payment confirmation).