Where do the names of the ecliptic and equator come from?

The name of the zodiac comes from the color of the sun.

The ecliptic is the orbit of the sun's annual apparent movement. The sun is yellow, hence its name.

The orbit of the moon (the projection on the celestial sphere) is called the white road, and so is it.

As for the equator, it should be because it receives the most sunlight and has a high temperature, and it is often represented in red on maps.