On the first video. the first attempt sounded like the starter momentarily engaged the engine and tried to crank then engine then it stopped. The subsequent attempts on both videos sounds like the starter is not engaging the engine. It is possible the piston or rings are stuck from sitting so long.

I'd probably disconnect the battery and pull the spark plugs. Put a cap full of motor oil or marvel mystery oil directly into each cylinder to coat the piston top and let it soak ito the rings. Try cranking the engine by hand with a 19mm socket at the crank bolt. You should be able to hand crank the engine and have consistent resistance and it won't take a ton of force. The torque needed would be the force required to loosen a moderatley tightened bolt, say the force required to initially loosen a caliper pin bolt. If you feel higher resistance in spots, there could be build up that is sticking the piston rings.

 

Before I put the new starter in I turned the fly wheel a little with a screw driver. It moved pretty easily. Then I installed the new starter. That's when It started to crank on two attempts to start it The second video is what happened after the next few attempts. It went back to making the whizzing sound with no crank. There was a small black box that had a wired connector attached to it. It was just hanging just under the dash near the obd port. It had a wire that went to a red button someone installed on the plastic panel just above the hood release. I'm going to reattach it and see if that does anything.

 

Looks like some kind of aftermarket security/remote/fuel cutoff setup. If you can find a part number, you can see how it is wired up and what it does. Since the starter did spin, I don't think this box is preventing signal to the starter.

You could have a bad section of battery cable from the + battery post to the large starter post, so you can't get enough current to engage the starter. A voltage drop test is easy to try. You need to hook up a volt meter on the battery + post and the starter large post. Have an assistant crank the engine, and look at the voltage. It should be <0.5V. Anything higher means the battery cable has a section of high resistance, like dirty/corroded connections that need cleaned, or corrosion along the cable restricting current.

You could try a jumper cable from battery + to the large battery post. Attach the starter side first. You must be 100% sure the jumper cable only touches the starter post and not any metal nearby or you will short circuit the batery directly to ground.

It is possble you got a bad starter out of the box. If you still have the old starter, try getting the gear/bendix freed and lubricated and install the original starter?

 

 

I checked the hot wire to the starter with a circuit tester. It stayed on when I tried to start the engine. I next check the small plug that plugs into the starter. It was off then came on when I tried to start the engine. The starter seems good. I don't know where to go next

 

I noticed the flashing key light in the dash stays on when I tried starting the car. Is that supposed to happen or do I have to look into that as a possible cause? I have another 2001 Accord and that key stops flashing before I start the car and stays off after it starts. What am I missing?

 

That's the black box and the harness that's hooked up to it. Where the heck does that harness go if it's not connected to the box?

 

that's the red button it looks to be connected to

 

From your video and these pics, you definitely have some aftermarket setup on your car that you can try to sort out. The FCCID stamp on that box means that it almost has to recieve some remote signal. I'll guess remote start or a security system. Maybe you can get a part number off the circuit board of that black box. You'll have to trace the obvious aftermarket wires to the OEM harness to figure out what was done.

The aftermarket wiring doesn't explain why the starter isn't performing and for me is a rabbit hole. You are the 12V signal for the starter to engage and the large wire should be directly connected to the battery. The starter should crank over the engine based on your testing. The engine may not run on its own with the aftermarket wiring, but troubleshooting will be a future step.

The starter is a very simple circuit. If the power cable from the battery doens't have high resistance, and you get the 12V signal on the small wire, then you could have a weak ground for the engine block. A simple bypass for ground is to connect a jumper cable to the body of the starter or a bolt holding the starter to the engine. Then put the other end of that jump on the negative battery post.