It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant").
It is noteworthy that many of the named cubic curves look rather similar: the folium of Descartes, the trisectrix of Maclaurin, the (right) strophoid, and the Tschirnhausen cubic look very similar in form; the semicubical parabola and the cissoid of Diocles resemble each other as well.
I have deliberately placed the word "kind" in quotes since there does not seem to me an intuitive way of defining the term, so an answer to my question might have to define "kind" rigorously in the context of cubic curves. (An algebraic invariant, for instance... it is a pity that there does not seem to be an analogue of "eccentricity" for cubics!)
In here, it is noted that Newton classified cubics into 72 "kinds", and Plücker after him described 219 "kinds".
So, how does one algebraically distinguish one cubic curve from another, and with a rigorous definition of "kind", how many cubics are there?