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I thought molecules that have a ring have an added degree of unsaturation, but this explanation has confused me:

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How can something like cyclohexane which has degree of unsaturation of 1, be considered to be "saturated"? Are there different meanings of saturation?

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Saturated is an old term in organic chemistry, dating back when the atomic structures and electrons did not exist. The meaning of saturated, even today is "chemically combined to the maximum extent, having no ability to combine further." The carbon atoms in cycloalkane are saturated in the sense that all the four valencies of each carbon atom are satisfied by four individual atoms.

The degree of unsaturation is a related concept: It tells us how many rings or double bonds are in the molecule. Don't mix these two terms.

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