As I understand it, the field coils and armature windings in a DC motor are insulated and this insulation can 'burn off' if the motor overheats, causing a short circuit to develop. What is the material coating the windings and coils in a DC motor? In the examples of the insides of DC motors that I have seen, the windings still retained the look of copper wire - does this mean that the coating is transparent? If so, is this always the case or are there winding coatings that are not transparent?
Wire used for motor windings, transformers and other electro-magnetic applications is called magnet wire. Magnet wire insulation is similar to polyurethane varnish. However is is developed specifically for electrical insulation use. As you have observed, the insulation is quite thin yet able to withstand the normal operating temperature and voltage required for motor windings. It is not completely transparent. It is sometimes colored green or other colors that do not look like copper. The most common color is similar to copper. There doesn't seem to be any particular reason to use other colors.
Various polymers are used depending on the temperature required. Look up 'Magnet wire' on Wikipedia. I haven't seen opaque 'enamel' used.