Most kill switches on older 2- strokes will be a "make to stop" - it short circuits the spark plugs to stop the engine
The switch on the end of the handle is simply a "push to make". Same theory applies to the kill switch, except it "defaults" shut and is held open by the presence of the driver who of course has attatched the lanyard to their anatomy..... The one on the handle is more of a convenience device. I have a similar button on the casing of my '72 Johnson 25.
You have two options, you can either wire the deadman switch on parallel, or you can disconnect the one on the handle & replace it (electrically) with the deadman switch. It really depends on a combination of do you want the "push to stop" convenience, or are youhappy stopping it by pulling the deadman, and how good you are with electrical crimps to make the Y- splitter on the wiring up!
That ebay one looks like it might be designed for twin engines, as the contacts appear to be duplicated
I've got a Clymer at home, I'll see if I can work out the wire colours for you tonight.