Open the hood, then looking down near the base of the steering column, you will see a lever at the end of the shift tube. There is a rubber grommet in that lever where the shift link from the transmission connects. When that rubber grommet gets soft or wears out, your gear shift lever will sag a little. When that happens, the pin at the other end of that lever moves the neutral switch contacts off the mark.
So in order to start the engine like this, you have to nudge the gear shift lever a little to get the lever to activate the neutral switch contacts.
Many years ago, the repro parts suppliers were selling the rubber grommet, nylon insert, washers and cotter pin in a kit for Fordomatics. But I haven't seen them in a long time. That rubber grommet needs to be stout to make the system work properly.