Starter is hanging up after vehicle starts and will not disengage if you unbolt the starter and roll it over the starter works properly what would cause it to hang up

In most cases it is the "Bendix" or the over-running clutch, depending on what the part is called by your mechanics, parts stores, and garages locally.
I really like the pictures you show.
In your picture, you show the "business end" of the starter, or that end which fits inside the transmission housing that you normally do not see when the starter is bolted properly in place.
If you will look at your picture, you will see a tiny gear about 1 & 1/2 inches in diameter. This is the problem part, except that this gear is a part of the part called the "Bendix" or the over-running clutch.
You will be needing to take the starter all apart on your workbench and replace that part, which should cost you 10-30 dollars.
That gear itself is not the problem, it is the spring behind it most likely, or the Bendix is perhaps sticking on the central shaft and not returning due to dirt or rust. You will see when you dismantle it.
When you get the starter apart, go ahead and "rebuild" it!
Start with the solenoid.
Carefully take it apart, and take a piece of fine sandpaper and clean the copper round plunger, about the shape of a mushroom. You might also gently clean the contacts inside that touch the plunger.
If your solenoid has soldered connectors, skip this step.
Now dismantle the starter itself, remembering the brushes require some fancy finagling to get them back on the armature! ( Like using some paper clips to hold the brushes in their pockets to get the armature back into position).
You should clean the copper part of the armature with that fine sandpaper. That part of the armature is called the commutator.
Be really gentle here, and replace the brushes if they are worn way down. If you have a car parts store available, ask to see new brushes, to tell how much yours are worn.
I recommend you pack the bronze bearing inside the end cap of the started with a half-thimble full of good grease, and also grease the end of the armature that fits into the bronze bushing on the "business end" that goes inside the transmission housing.
(Have a good look at both the bronze bushings, to be sure neither is badly worn. The also are replaceable for a few $s.)