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That's an odd one. Given the symptoms, my guess is a weak electrical component. Since you have no noise whatsoever, perhaps it is the engine starter motor relay in the power distribution center. When the car is acting up you can pop the relay and try a quick jumper (with the car in Neutral!!!) to see if the starter solenoid will engage.
You need to locate the battery lead connector and the relay output connector (The relay has 5 sockets under it on my 96.) On mine, the battery lead is at the 9 o'clock position, and the relay output is at the 3 o'clock position. If you can remove the relay and get the solenoid to engage by jumping while the car is balking, then the relay is the culprit. Just be careful not to run over yourself.
Since the starter is new, is this an ongoing problem that a new starter did not fix, or an unrelated one?
Also, poke around near the starter itself. Look for loose or dirty connections. Heat raises electrical resistance so a marginal connection can turn into a failing one after a little heat soak. Been there and done that on a brand-x car before.
Good luck, this one sounds tricky!
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