Mine is a 97, and it may be different from the 02 Neons. Mine only has one fusible link, located at the starter motor end of the wire that comes from the alternator. If it burned out, it would only prevent the alternator from charging the car (which is done via the heavy cable from the starter back to the battery). It wouldnt have any affect on the starter working.
Under the hood and adjoining the battery I think your car has a fuse and relay box. Mine has a 30A fuse for the starter and a starter relay in there. All of the several relays are the same, so I can swap around to check them out; for instance, swapping the starter relay for the horn relay and seeing if the horn still beeps. But I think you already checked the relay. The 30A fuse has metal contacts that I believe can be reached with the pointed tips of probes from a voltmeter. For sure there would need to be 12 volts on that fuse when the ign. key was turned to the 'start' position.
My guess would still be that the neutral safety switch connector was loose or damaged.