I think the lead in post #2 is used most often for the supply to a satnav, most commonly on 1200GS's connected to the blanked-off supply hidden awaynear the oil cooler / beak.
The lead that taps into the supply beneath the seat is / was a much heavier and longer BMW extension lead, used to take a supply from under the seat up to the (optional) socket on 1200GS bikes on the near side front fairing.
I think there are two sorts, as I vaguely remember that BMW changed the design of the under seat socket's supply.
If you just want to trigger your relay, the lead in post #2 will probably do, even so.