Yes that adapter and switch match up pin out wise. Keyed area matches the switch shown. You splice it in using standard butt connectors, just be carefully and not cross any of the wires. you also may have at least one unused wire. You toss the un-used end when you are done. I know it is a waste on materials but as a non shop installer that would be you best solution. The key connector housing makes sure that it and switch goes only way together.
The housing shown is the same one JD uses (JD PN M117518) and the terminals inside it are the Packard 2965867 (16-14 gauge). Of course the wire is color coded. As I said using a standard crimper on these F56 terminal only results in a loose crimp as they are open barrel crimp style terminals. So far JD is only place I have found for either of the particular keyed housings. Since I keep both versions in stock it isn't a problem for me to go to electrical drawer for whichever one I need, There is one note I should share here about those F56 females terminals which is the spring hinged contact does tend to break at times and causes a loose contact which leads to overheating the connection. This sounds like what be happening on your end since the PTO clutch resistance is within specs.
Just make sure your switch is keyed as shown as there is one other style of keyed version where the keyed area is is one side of the last row and not in the center, I would have looked up your mower but I needed the actual model number from the serial number tag.