The definitions of left- and right-handed coordinate systems never clear anything up for me. Following is my current understanding of how the systems are determined and why they are given their names.
A right-handed coordinate system is one that can be modeled on your right hand. Point your index finger along the positive x-axis ([1 0 0]). Point your middle finger along the viewing direction. When you take the cross product of your index finger and the viewing direction, you get the positive y-axis ([0 1 0]). For index-cross-middle to produce thumb, the direction the middle finger points must be along the negative z-axis ([0 0 -1]). This is easily proven with your favorite cross product calculator.
In other words, we can define a right-handed coordinate system as one where taking the cross product of [1 0 0] with the viewing direction produces [0 1 0]. The only viewing direction that allows this and is orthogonal to both these vectors is [0 0 -1].
A left-handed coordinate system can be defined as one where taking the cross product of the viewing direction and [1 0 0] produces [0 1 0]. (Note the order has flipped.) The only viewing direction that allows this and is orthogonal to both these vectors is [0 0 1].