Chinese Embassy website information is often incorrect and/or outdated. The specific current Chinese regulation, which is translated correctly in the Timatic database, explicitly allows leaving the airport while on visa-free transit. However, as has been said above and is in the FAQ Sticky on the China forum, CAN (only) may be experimenting this summer with a procedure that will not allow leaving the airport unless one has a valid Chinese visa. Not enough data points at this time to confirm or deny a diverging CAN policy, though.

Picking up luggage while on transit-without-visa is subject to individual airport policy and procedures, not central government and not airilne policy. At PEK for instance, intl transit passengers need not pick up luggage and recheck; as long as it's through-tagged to ultimate destination, the baggage handling system will deal with. But passengers and their carryons still have to go through immigration then another round of security. At PVG however, policy requires intl transit passengers to pick up bags and recheck onward, as well as do the immigration procedure. I haven't done CAN transit since 2009, but it was an auto-check-through/no manual pickup similar to PEK back then, and I suspect hasn't changed.