From the point of view of Fractal Art rather than Fractal Math the definition of Mandelbrots and Julias is a lot looser - specifically a Mandelbrot is simply considered as any render where the orbits used to produce the image all start with a fixed value of z and some constant in the iterated formula (c) is taken from the position, whereas for Julias the start value of z is taken from the position and the constant (c) is fixed for a particular Julia Set.
In other words a Mandelbrot (as in most fractal programs) is the result of iterating a related set of different functions (generally the functions differing by the variation of the single constant c) and a Julia Set (again as in most fractal programs) is the result of iterating a single function just with different starting points.
Whether the functions involving c are rational, irrational, transcendental, use conditionals etc. makes no real difference provided one remembers that some attractors/attractor sets are at infinity, some single fixed points, some periodic (set of points), some non-periodic but finite (that may or may not be connected e.g. strange attractors) and some (that fit into none of the previous categories) simply an infinite set (that also may or may not be connected).
The last one is *very* difficult to distinguish from the others for instance this attractor (a single orbit) looks like a strange attractor even after many, many iterations but does gradually drift possibly even being divergent in the end....
http://makinmagic.deviantart.com/art/The-Infinite-Harp-40419447As for formulas that reproduce copies of Mandelbrots, my personal favourites are the classic Magnet formulas and one of my own that came about by trying (against all reason) to find the true half-iterate formula for z^4+c which ended up as "The Magic Formula" in mmf3.ufm for Ultra Fractal (in the Ultra Fractal formula database -
http://formulas.ultrafractal.com/ ) in which for certain versions of the formula (as set by the user parameters) there are n^i minibrots within each iteration band starting with the second (where i is either iteration or iteration-1 I forget which) - each being at the centre of a disconnected mini-Julia that matches the true Julia shape from around that location on the Mandelbrot.
If interested then see the comments at the start of "The Magic Formula" in the ufm text file.