You must have the "A" or "L" model, then 817778A-1 is the correct part number for the thermoswitch. The other part number is also thermoswitch but mostly for the 125's, 90's and 50/55 HP. F629449-2 is for the "B", "C", "M" and "N" models, 86-85HP.
Now the final question is, if the thermoswitch for the 125 is interchangeable with your 86-85HP. That I have no idea. But looking at the description and pics on eb@y it looks like the switch itself is the same. The 817939A1 seems to be a kit while the other is just the thermoswitch itself.
upon further review, it looks like the 939a1 switch has F170 printed on it, the 778a1 switch has F180 printed on it, could this be the activation temp? I have no idea, sure sounds reasonable....