Correct on the different starters. Though it's not a tooth count issue, as that's taken up by the bell housing. It's indeed the auto-vs-manual trans issue.
There is actually at least one more starter style for our engines, but it's different enough that it's obvious at the mounting flange.
Yes, they make shims nowadays for Ford starters, but it's not as common an issue as it used to be on GM engines. However if it's not engaging the ring gear now, shimming it out could make it worse if the problem is that it's already not reaching.
Check out the different pictures comparing the nose cones and gear locations of the two styles and you can compare to what you have.
Is your '73 a manual or automatic truck?
Paul