1. "I did not see where JMB called the 1/2 cock notch specifically as the "safety position."
It's there. Keep lookin'.
2. "He did say "lower the hammer to the safety position," but nowhere does he specify that to be the 1/2 cock notch."
It states:
"To lower the hammer to the safety position without touching the firing pin."
Since the only place the hammer touches the firing pin is full down, that only leaves the half-cock position.
Note that I'm not advising you to use it as a safe carry mode. Only alluding to Browning's intent. It is a de facto safety.
3. When the hammer is in half/quarter/safety cock whether with a captive or not the trigger is limited in travel front to back.
In my experience, when the sear is captive in an original design half cock notch, the trigger is securely frozen in place with, at most, barely detectable front to rear movement...and only if manipulated with some force.
All my pistols, save the two Colt 1991A1s that I use for beaters...numbering nearly 4 dozen...have captive half cock notches on the hammers. Only one lets the trigger move front to rear by more than about the thickness of a sheet of paper. One bone stock Norinco gives up about a 64th inch of travel...and the trigger in that one sits about that far below flush with the rear frame wall.
My first suggestion to the OP is to determine which hammer is in his gun. Unless he bought it spankin' new, there's a chance that somebody swapped it out for a Series 80 hammer. If it's original or correct, either the half cock notch has been damaged or modified, or something is keeping the sear from fully resetting and it's staging on the edge...which is unlikely because if the sear won't reset sufficiently to grab the half-cock...it probably won't engage the hooks and hold at full cock.