Its definitely the solonoid not engaging and spinning. Like it extends but is blocked and spins instead of engaging. Its the same kind of sound when teeth are missing or an old starter isn't extending enough.
This happended to a '98 Expedition with 5.4L, but only screeches once maybe every 2 months. That one is the same brand, which is why I suspect how I put it in stead of missing flywheel teeth. Either is possible.
THIS '99 has 2 bolts INSTEAD of fully threaded holes at the two starter bolts had to be drilled out. But sume threads on THIS '99 Expedition we are mainly talking about are still there. So 2 longer than normal starter bolts are used in the lower 2 positions until relatively tight, then a lock washer and nut are used on the back side. The bolts and nuts came from another engine are are strong and hardened.
UPDATE: Just tried to start it and it fired right up. When it started screeching, it was the day after I put the starter in. After I drove it for 30 mins, and parked it for 1 min, that was the first screech. Then I ran it for another 20 mins. Parked it for 15 mins, and then when I tried to start it again, screeched once then started. Not sure if it's related to heat or luck. I am reluctant to drive it anywhere until I think about re-adjusting the starter, but not sure which way to tilt it if anywhere. When I put it in, I just hung it off the top bolt (which was intact and you have to put in blind), then connected the bottom 2 bolt and nuts.