Some writers seemed to distinguish between a Greek amphitheater and a Roman amphitheater, with the Roman amphitheater dated to before Rome took over Syracuse.

This may have been an issue of hypercorrection. The people saying amphitheater probably thought it was to be preferred to theater. Perhaps they thought it was right in the context of an outdoor place or an area that fell into Roman hands. Actually, I don't know. What I do know is that I was confused. And I may still be missing something.
While I'm not certain I've ironed out all the details, I have made an attempt to provide background on some of the sites of Syracuse, Sicily, still visible today. See Remains of Ancient Greco-Roman Syracuse.
As always, please let me know if I've made mistakes.
Photo of the Theater CC Flickr User Alun Salt.

Comments
This sent me running to my OED, which gives its 6th meaning for amphitheatre as a semicircular rising gallery in a theatre (M19). Therefore a Greek theatre is by modern usage an amphitheatre. (Yes, I know amphi in Greek means enclosed circular.)
Theatre in modern, popular usage has a clear implication of seats in a rectangular formation, inside a rectangular prism, facing a reectangular stage. I can see why Greek theatres have attracted the name amphitheatre.
Gary, I was missing the rectangular bit. With that it makes better sense. Thank you.