There are more than two, Unreal has a coordinate system so special that Tim Sweeney apologized for it.

https://mobile.twitter.com/timsweeneyepic/status/95266147450...

There are also left-handed and right-handed coordinate systems...

This is correct. Most CG and 3d software (Max, Maya, Blender, CAD) uses a right handed coordinate system, which matches the convention in physics, however they vary in their choice of Y up or Z up. Video games almost always use a left-handed coordinate system with Y up. This is because DirectX and OpenGL chose to represent pixels on-screen with (X,Y) coordinates with the Z coordinate increasing as rays extend from the eye outward.

Wow, I had no idea it was so varied

To add to the confusion, while in both OpenGL and DirectX treat Z axis as depth, DirectX uses a left-handed coordinate system, while OpenGL uses mostly[0] right-handed one.

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[0] - https://stackoverflow.com/questions/4124041/is-opengl-coordi...

OpenGL-the-interface is kinda neither (screenspace aside), since the application supplies/does all transformations (glFrustum and friends are long deprecated).