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You could try Tglad's system. It differentiates in a different way than you propose, but it might be what you are looking for. It must be somewhere here on the forums...

Attractors? There are lots of them indeed.

A list can be: http://www.bentamari.com/attractors.html

In your list you forgot all escape time classic fractals? Mandelbrot and Julia modes... Nevermind I did not read

I like to categorize fractals by how they are defined. The most common I've seen are:

* Recursive geometric operations
** L-systems
** iterative deletions (e.g., Cantor set, Sierpinski gasket, Menger sponge)

* Random fractals
** DLA
** fBm

* Orbital fractals
** strange attractors (e.g. Lorentz, Hénon, Rössler)
** IFS
** fractal flames

* Escape-time fractals
** Julia sets
** Mandelbrot sets
** Mandelbulbs

The last two both use orbits of points obtained by repeated iteration of functions. With Orbital fractals, the fractal consists of the orbit points; with Escape-time fractals, the fractal consists of points whose orbit is bounded. The term "escape-time" is a bit of a misnomer since it applies to a common computer algorithm for generating images of such fractals, where points that are not in the fractal (they "escape") are colored by the length of the pre-escape orbit. Mathematically, points whose orbit is unbounded are simply not in the fractal.

One disadvantage of this categorization is that many fractals can be defined in different ways. For example, the Sierpinski gasket is easily defined using recursive geometric operations, by an IFS, or as an escape-time fractal. But perhaps this isn't too bad since computer-generated approximations using different methods are subtly different, even if the mathematical object is the same in all cases.

welcome to the forum &  thanks for the input!
into which category would you sort these two? both escape time fractals?