We don't "know" how gas giants form, but it is a good bet that there is some variation from gas giant to gas giant. The scenario I imagine to be the case is this: Some begin as your type two, but the process stops at some point when gas gets blown out of the proto-solar system from a nearby supernova, or onset of fusion in the most massive core in the system. Then a dusty disk forms around the new star and accretion of the dust commences.
If this is how it works, I would guess that most of Jupiter's mass is from your type 2, and some from type 1. Saturn is closer to an even proportion, and the other planets are mostly type 1 accretion.
Forming opinions as we speak