That is quite an old article.
On your original wiring you had 2 wires off the alternator.
A red wire with a diode in it to make it DC ( thick bit near the plug ) and an AC black wire which went to the lights.
The new engine has a DC only output cause it has the real rectifier on it in place of the single diode.
So the red from the rectifier goes to the red wire on the old plug hiding behind the dip stick tube.
For the time being you can forget the old black ( AC ) wire as it only goes to the headlamps and to get them to work might require a bit more wiring.
Then you only need to hook up the magneto kill wire and the fuel solenoid wire .
The grey wire on the new 2 pin plug should be the solenoid wire and the black wire should be magneto kill wire, remove them all together as you will be using the old wires.
Run the old black magneto wire up to the magneto and the old solenoid wire to the solenoid, they should go behind the starter not in front of it.
Now cut the 2 wire plug attached to the old 4 pin plug off near the 2 pin plug so you have some wire to work with .
Cut the white plug off the red wire coming out of the rectifier get a single connector plug and join the 2 red wires together using a plug of some sort .
Resist the temptation to solder them together because if the rectifier fails you will have to cut the wires again & eventually they get too short to work with.
From here you are good to mow.
This will only leave you with the lights ( if you want them ) and for them you have to do some rewiring as they are always in circuit ( you turn them on by grounding them ) and can either drain the battery or worse pump power to the magneto when the mower is turned off.
To work out how to wire them in I need to have a look at the connections on your ignition switch.