No, I’m referring to the Iridium pager, which is just its own service and has been around longer than SBD, I believe. Messages were (are?) free to send on top of a monthly flat subscription rate per pager.
GDB is just a branding for their own flavor of what they're selling which is Iridium Burst, which also appear to run on the 9602/9603 SBD modules, so... maybe Iridium Burst is just spicier Iridium SBD?
SBD is bidirectional unicast (and mandatorily so; as you say, SBD can't transmit to devices before they initiate a mailbox exchange, with the exception of a "message waiting" indicator.
Iridium Burst seems to be multicast-capable, and optionally bidirectional or unidirectional, so "spicier SBD" makes sense!
GDB seems to be a successor to that, though.