The correct botanical genus name is Bucida, and apparently when I wasn't paying attention there was a revision, no longer Bucida spinosa, but now called Bucida molinetii, I would change the tag on your tree to read Bucida. I'd probably keep calling it Bucida spinosa until I saw more references adopt the revised name of Bucida molinetii. Botanical taxonomy works by consensus, a name is proposed, and whether or not it gets adopted by several key organizations determines whether the change sticks or not. "
Nomen nudum" is the fate of discarded names, and the name of the circle of hell reserved for bad taxonomists.
Bucida spinosa is in the leadwood tree family,
Combretaceae, not at all related to the Mediterranean edible olive, genus Olea in the Oleaceae - the whole family with several thousand species was named for the edible olive. So the spiny black olive is not at all related to the edible olive. Completely different evolutionary tract, completely different families.
Myself, I would not chop it low. As in my original post, I did say, choose one of the branches in the cluster that you identified. My reason is that Bucida is a true tropical tree, you are growing it out of its natural range, which means it grows slower than normal. The low chops would get rid of relatively old wood, that would require years to grow a replacement. The Bucida I've seen grown in my Chicago - Milwaukee area climate, they really only have one or two months a year warm enough to get any active growth, first month of Wisconsin summer is lost just getting the tree to wake up, then next month is what little growth you'll get and by then it is September, where one cold night can slow or stop all that great growth. The rest of the time they seem to sit semi-dormant. They love warm temperatures. This is also the reason Buttonwood mangroves do not do well in the midwest. They too are very tropical trees.
So in Tallahassee you do have more of a suitable climate for Bucida than Milwaukee. You probably have 8 or more months where it will be able to grow. Which is why I asked you does it grow fast or slow? If it is slow growing for you then choose on of the larger branches, if it grows fast for you, then choose the thinnest.
I really would not chop it low, despite my esteemed colleagues suggestions, I don't believe they have much experience with the species. I know I don't have experience. I would definitely get to the Tallahassee Bonsai Society meetings and see if someone there has good experience with Bucida. Use the botanical genus name, so you don't end up ''down the rabbit hole'' talking with people thinking Mediterranean edible olive.