After some very early starters that IIRC went away by 1962, slant six starters are all the same (and the same as V8s from the same era).
There are some asterisks, terms and conditions on that. The direct-drive '60-'61 starter you're thinking of (with 9-tooth drive pinion) is not interchangeable with the '62-up gear-reduction starter (with 10-tooth drive pinion). But the direct-drive, 9-tooth starter was used on Canadian-built Slant-6s through '66, and on some US-built '63 Slant-6s in cars, and on some US-built pre-'73 Slant-6 Dodge trucks, and there have been shadowy reports of their use on some '70-'71 US-built Slant-6 cars.
And it would take some serious creativity to get a wrong-sized ring gear on there.
No creativity required. The two applicable ring gears are as follows:
Chrysler p/n 2121 195 (ATP ZA-545): 148 teeth, for use with 9-tooth starter
Chrysler p/n 2121 196 (ATP ZA-505): 122 teeth, for use with 10-tooth starter
All that said, if this car hasn't had a starter or transmission swap, the problem is probably just a faulty starter. (On the other-other hand, maybe the car was parked long-long ago because someone did a parts swap and couldn't get the starter to work and gave up…!)
But you say "starter doesn't look old". Maybe show us a pic of the starter; let's see what kind it is. Judging from pics in your quick-flip thread, it's a US car, not a Canadian unit.
Er-ruhh…you did check your battery cables to make sure they're not internally broken, and the battery is known to be good, yeh?