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?
The most standard notion of “kind” is that of isomorphism in algebraic geometry, that is, there are polynomial maps from one curve to the other which, when composed, are the identity on the curves (they don’t have to be the identity in other places).
For real, affine plane curves, this notion reproduces the fact that there is a unique line and exactly three smooth conics (over the complex numbers, ellipses and hyperbolas turn out to be the same, and in the projective plane, all are the same).
However, regardless of ground field (well, regardless of working over the real or complex numbers), there are infinitely many different cubics up to isomorphism, determined by the j-invariant, one for every element of the field.
A conclusion you can draw from this is that there really isn’t a good algebraic way to say that there are only finitely many types of plane cubic, unless you want to lump all smooth cubics together (which you clearly don’t, as you consider parabolas and hyperbolas and ellipses to be distinct). I don’t know what method Newton and Plücker used to classify them, but isomorphism is the standard method in modern algebraic geometry.
In John Stillwell’s Mathematics and its History it is observed that Euler criticized Newton’s classification for lacking a general principle, but that a closer examination of Newton/s work reveals one. In fact his work gives a general classification into 5 types, depicted on page 112 of that book.
I believe the difference in the real and complex cases, is the fact that the “discriminant” locus of singular cubics has real codimension one in the real case, hence separates the space of smooth cubics into distinct connected components, and this may be the types sought for. By Ehresmann’s theorem, a connected family of compact manifolds have the same topological type. Thus curves on different connected components of the complement of the discriminant locus can have different homeomorphism type.
At least one (two?) of Newton’s types is also singular and could represent the general (and special) point of the discriminant locus. I am not an expert.