"Heat of hydrogenation" is a subset of "heat of reaction" which is nothing but the standard enthalpy change ($\Delta H^{\circ}$) in the system as reactants are converted to products. In catalytic hydrogenation, as hydrogen molecules are added to atoms bearing double/triple bonds, the enthalpy change which accompanies the reaction is called as "heat of hydrogenation".
Now coming to your question:
Does the heat of hydrogenation include only the double bonds in an alkene or does it also include other forms of unsaturation, such as rings?
Heat of hydrogenation just refers to the enthalpy change required to convert double and triple bonds to single bonds in a molecule. It doesn't account for ring opening. This is because opening a ring involves breaking at least one carbon-carbon covalent bond which requires a great deal of energy. But that doesn't mean rings cannot be broken. If they could be broken, then the enthalpy change is plainly called as "heat of reaction" but not "heat of hydrogenation" since it would lead to ambiguities.
Thus, heat of hydrogenation of cyclooctatetraene is just the enthalpy change associated with those four double bonds, even though cyclooctane has a Index of hydrogen deficiency (degree of unsaturation) as one.