You're confusing Type III with Type V (of the infamous 5 over 1) construction. Type III construction uses a non-combustible exterior material like brick and stone around fire-retardant-treated exterior wooden framing members. The A vs B is whether or not the interior framing members need flame-retardant. B means they do not. This is typically allowed when the exterior material is very fire-resistant, brick, stone, etc, as opposed to say steel or glass. Type V on the other hand is combustible material "with low or no fire-resistant construction." These have the lowest occupancy and height limits and require sprinkler systems even if only 2 stories. Type V allows whatever the cheapest thing you can build that is up to code (which is plywood in the US) whilst Type IIIB requires some kind of fire-proofness built in.
In addition, dont be perturbed by the term "glulam" lumber. It's different from plywood and is legitimate wooden framing members, but reinforced with glue, making them stronger, capable of more complex shapes, and longer lasting than plan lumber.