Car won't start unless I push it.

You're referring to the neutral safety switch, but that is only used with automatic transmissions.

The first problem is if you removed the starter and generator to take them for testing, that is very inaccurate. A starter has to be under the load of trying to spin the engine when it's being tested, otherwise a weak starter can appear to be okay. This also makes a real lot more work for you. A good description of the symptoms, then a few quick checks should identify the cause of the problem, but since this started out as an intermittent one, the goal now is to keep it that way so there is a defect to be found. A lot of people poke and wiggle until the circuit starts working, then there's no point in testing anything because everything will test good.

Based on your dandy observation that wiggling the shifter affected the symptom, I'm going to guess that when this doesn't work, you don't hear any sound from under the hood when you turn the ignition switch to "crank". If that is wrong, let me know.

Ford used at least three versions of this circuit, sometimes I think just to confuse everyone. The solenoids are different, and while some can be interchanged if you know which terminal to use, it will be best if you put the original one back in.

There should only be one small terminal on it. A red / blue wire connects to it. To verify everything else is working, shift the transmission to neutral, because you'll look funny chasing after your car in the next step! Leave the ignition switch off. Remove that red / blue wire, then use a piece of wire to jump from the battery's positive post to that small terminal on the relay where you took the red / blue wire off of. The solenoid, (relay), should click very loudly and the starter should crank the engine. If it does not, we have to figure out if you reconnected something wrong.

If that works, there are only two things in that circuit, besides connections, that can cause this. One is the ignition switch and the other is the clutch switch. Neither has anything to do with wiggling the shifter, so I'm leaning toward a bad clutch switch.

Ford used a pretty sad design for their ignition switches too. There's two ways to approach this. The first is to assume the clutch switch is defective. Crawl under the dash and look for it. It will have two white / pink wires. Don't bother trying to test it. Unplug it, then use a stretched-out paper clip to jump the two terminals together on the harness part of the connector. That will bypass the switch so you'll be able to crank the engine without pushing the clutch pedal. If you can start the engine now, replace the clutch switch.

If it still doesn't crank, we need to measure the voltage on that paper clip. The easiest is to use a test light. Connect the clip on the test light's wire to that paper clip, then touch the probe to a paint-free point on the body, or an unpainted bolt head. This is actually connecting it backward but that doesn't matter. This lets you do the next step without needing a helper. Turn the ignition switch to "crank". If the test light lights up, the ignition switch is okay. If it doesn't, we'll have to dig into the ignition switch.