In the case of quadratic curves it is widely known that a conic is uniquely determined by 5 points. But if we use only three points we can uniquely determine a circle and if four points -- a parabola. Both are special curves in some sense (the first one is special in Euclidean geometry and the other one is special in Affine/Projective geometry).
The question is probably a little bit vague, but what special types of cubics can be determined by just 6 or 7 points?
A circle is a conic through the two ideal circle points $[1:\pm i:0]$, so this is just another case of a conic through five points, with two chosen in a special way. You could simply generalize this to cubics by taking these ideal circle points as points on the cubic as well. Since they play a very central part in modelling Euclidean geometry in the projective plane, the result will likely have interesting Euclidean properties as well, but I don't know which.
A parabola is a conic tangent to the line at infinity. So if you want to define a parabola by four points, you are in a special case of the more generic problem of finding a conic given four points and one tangent line. Which has two solutions in general, and even in your specific situation.
If you want things to be uniquely determined, you should define a parabola by four tangent lines. This you can again generalize to the cubic case, by taking the line at infinity and 7 more tangents. Or you stick with 7 points and one tangent line at infinity, and accept that the results are no longer unique.
