So, let's just cover a little general theory first so we are on the same page.
A starter will have two wires connected, one just a regular looking wire, and one heavier cable. The regular wire is, in essence a trigger wire. Turning the key to START applies voltage to this wire via the fender-mounted starter relay. Applying power to this regular wire at the starter energizes the starter-mounted solenoid, which then mechanically pushes the bendix out to engage the fly-wheel.
Now, when the bendix is fully extended, that is, fully extended and engaging the flywheel, the starter-mounted solenoid closes a set of internal contacts. You can't see these contacts, they are inside the solenoid under the bakelite plastic end cap where the wires connect.
(trivia break: Did you know that, at one time, people used to replace just these contacts rather than the entire starter? You could buy a little kit for $5.00. That was before everything became disposable)
Ok, now I have to go back to the heavier cable. The heavy cable connects directly to the battery (+). The internal contacts connect this power to the starter motor itself.
So... turn key, fender-mounted relay engages, starter solenoid engages, bendix extends, contacts close, starter turns, car starts.
There is a method to this madness. You do NOT want the starter to turn until you know the bendix is engaged, otherwise you can trash the starter and flywheel.
With this understood, what is going on in this case?
It sounds like the bendix is engaging, or at least trying to. It would be good to verify this by opening the hood and listening for where the sound is coming from. But, my old ears are hearing some mechanical noise, so I am assuming this to be the case, that the bendix is at least trying to engage.
With this assumption, here are some possible causes for your problem:
1) A poor connection somewhere on the heavy cable. For this, I would start by looking at the starter where the heavy cable connects. It's a big lug on a big threaded stud with a nut. This connection can be ok and then go bad. What happens is, there might be a little film of grease on the wire lug and over time, the jolting starter, the engine vibration, this connection loosens up slightly. This connection has to be really good, otherwise power never gets to the contacts so the starter never turns. Take this connection apart after disconnecting the battery of course, and use some degreaser on the lug, the threaded stud, and the threads inside the nut. Then put it back together and tighten it up as tight as you dare. It will take some torque, but use a good wrench, you don't want to round the flats on the nut... the nut is soft metal.
2) Something has happened to the starter mounting where the bendix is no longer lined up to engage the flywheel properly. This can be as simple as a starter mounting bolt backing out. Again, a little grease on the threads, the jolt and vibration can back out a bolt. This gets a little more common with older engines because the bolts thread into the bell housing which, if I recall, is aluminum, so the threads can get a little wonky after the starter gets swapped out for the 10th time. I had this exact thing happen on my explorer where the bell housing threads went bad. As I recall, I ended up using a nut on the other side. A bit of a hack.
3) Power is not getting on the heavy cable on the battery end. Inspect the (+) clamp, peel back the insulation on the cable a little bit and inspect for corrosion. After you do this, apply a little grease or something to the wire you exposed, otherwise you will eventually get that corrosion... and you do not want to have to change that cable.
4) Defective starter.
I guess if I had this job, I would look at the battery connection first because it's a one-minute thing. If that was ok, I'd have to get access to the starter and check the other things, and if everything looked ok, I would get the starter tested.
) no disrespect. Sometimes one is focused on that one thing...
