atmosphere - What causes the aurora of other planets to have different colors?

Ultimately, you are correct. However, there may also be some clarification in order: what blue aurora of Saturn are you referring to? The most well-known auroral images of Saturn like this one, are not true color images. That is, what we see in the photo isn't what you'd see with the naked eye; instead, the photo has been enhanced to emphasize something interesting, in this case the aurora. In reality, that beautiful blue you see in the picture isn't even of light that's visible with our eyes. It's UV light which would be completely invisible to us. People working with data taken by a specialized camera have to give the UV light some visible color or we wouldn't see it at all. The Saturn aurora images I am aware of all have been modified in this manner.

However, aurora themselves are the result of charged particles funneling down a planet's magnetic field and crashing into the planet's atmosphere. On Earth, the aurora occurs because of the atoms in the atmosphere getting energized by these collisions, and then emitting light when they de-energize. Here, it's the oxygen and nitrogen in the atmosphere that are responsible for aurora light (primarily greens from oxygen, but also reds at higher altitudes from oxygen and occasionally blues from nitrogen). In other planets' atmospheres dominated by different atoms and molecules, the reactions will be different, and the wavelengths of light emitted will be different as well. For example, Saturn's aurora are due to energized H2 molecules. On Jupiter, it can be the infalling particles themselves (sulfur and oxygen) that are responsible for auroral emission.