I'd not heard of Pride of Barbados, so I looked it up. It's also known as dwarf poinciana, and there are a couple older examples around the forums if you use the search function. Using that name or the scientific botanical name caesalpinia pulcherrima you'll likely find plenty others online.
To answer your question, though, not necessarily. While technically the word, "bonsai," translates directly from Japanese to something like, "potted tree," it's not really used that simply. Generally it's expected to be stylized, usually to resemble an old full-sized tree of some sort. In fact, in the original Chinese tradition there's the practice of penjing which is the creation of whole landscapes in miniature using tiny trees. This style has been taken to its extreme in Vietnam with some works being bigger than a large pickup truck.
All that said, you're well on your way to making it a real bonsai, so keep going!
You can learn allot digging around this site, asking questions.