The key is in the free body diagram. With the Type 1, the rope/cable is in tension equivalent to the weight being hoisted - and if the winch side is approaching vertical, then the reaction at the pulley is 2F.
With the Type 2, presumably the chain/ratchet hoist is geared, likely with a greater than 10:1 ratio. Thus, someone pulling on the chainfall will exert less than 1/10 of the weight being lifted. To be more exact, perhaps the formula should be modified to account for the load on the chainfall side... but in reality, we're talking about a few percent so the load is considered neglegible and the simplification is made.
This is a good example of what can go wrong: A designer states on his drawing that the davit is a "Type 2" and designs away. The davit is installed in the field and the maintenance crew rig it up with a ground based winch - do you really expect the maintenance crew to a) have the drawing, b) review the drawing, and c) look up "Type 2" in Moss' book?
I'd design for a Type 1 in every case. Its much harder to mis-rig it in the field in a manner which would cause a problem.
jt