How many different parametric cubic curves are there? 2. The "same" game

I'm interested in both the answer to the question in the title and in finding the clearest possible derivation of that answer. My first efforts were very algebra-intensive, but had an aura of concreteness about them. As I played with the question, I came up with more and more elegant ways to ward off the brute-force algebra and make the answer more intuitive without reams of calculation. For these elegant techniques to work, however, I needed to build up a collection of tools (lemmas) that themselves take a little time to explain. One tool is the use of a particular matrix identity, which I review in this article, and which I use to answer the title question: there are three different parametric cubic curves. However, deriving that answer in a clear and obvious manner is still a little way away, but that's OK: the reason I'm doing this is really not so much to answer the question as to build intuition about the algebraic formulas and the geometry that they represent. The journey is the reward.