A fan dipole is usually constructed of a small-ish number (3 is typical) of thin wires, each trimmed to a specific length to have three distinct resonant modes and 3 distinct frequencies. Usually they are of substantially different lengths so each resonance falls on a different band. For example a fan dipole might be designed for 20, 17, and 15 meters.
A cage dipole is more like a single, very fat dipole with the insides of the conductors removed. All of it is about the same length, so we can expect one resonance where the dipole is 1/2 the wavelength, and at odd multiples thereof. However the resonance is wider than that of a thin dipole, since the resonant nodes aren't confined to such a thin line.
There are no hard and fast rules with antenna design. There are certainly possible designs which share characteristics of each of these designs. We simply give names to established categories, much like we do with art, such as "cubism" and "impressionism".
A turnstile antenna is quite a different beast. It is really two dipoles, each fed with a 90 degree phase shift. Such a thing can't be obtained by simply taking a fan dipole and spreading the elements farther apart. Obtaining the phase shift requires a particular feed arrangement, such as a power splitter with a 90 degree delay line on one of the branches.