There is a (de facto?) color standard for painted toroids. Micrometals appears to be the trend-setter here. Their products are bi-colored, with the first color covering three sides. Listings:
There may be exceptions; these are merely the most common cases.
Yellow-white is #26, a μr = 75 material with fairly high losses, typically used for filter inductors. It's a common sight in desktop PC power supplies, along with #52 (green-blue) which is almost identical with somewhat lower losses.
Solid colors are less well defined.
Black ferrite (coated, not bare) is usually a good choice for transformers, but beware as black is also used by Magnetics Inc.'s Kool-Mu® series, and maybe others.
Blue may be an expensive Permalloy (MPP) powder type, that is more efficient than powdered iron. See also Micrometals' Sendust material.
Green may be ferrite. White has also been used for both powder and ferrite materials.
PC power supplies can put out over 1,000 watts and they use E cores as they are easy to wind by machine. Large toroids need expensive winding machine heads so toroids are better used at low voltages were the number of turns is low, such as automotive amplifier power supplies.
If you are salvaging materials, a measurement of inductivity (AL), saturation current and Q factor are the best indicators of material type and suitability.
NOTE: Sometimes practical reasons determine what material and shape of transformer are used, which is not always the best choice. Cost and size compete with efficiency. The opinions of engineering and marketing and sales are not the same, and who wins determines what is used. "Just good enough" wins most of the time.
To cover all the variations of cores made by many manufactures you would need a book full of charts specific to each core material. For any given core of any shape you need the manufactures datasheet or chart for that core to get an idea of permeability and any hysteresis factors and peak current values vs. pulse width. To quote Ali..chenski's comment:
Ferromagnetic materials are characterized by more than one parameter,
by shape (and corner values) of its hysteresis loop, saturation field,
losses at various frequencies, etc. You can't map this
multidimensional parametric space onto a single color-coded lineup,
every material has variety of properties, and there are hundreds of
specific material compositions designed for different uses. Without
datasheets a core can be anything.
Link to Magnetic Hysteresis