Parts of the answer are in the text that you provide yourself. But I shall try to add where i can.
What do each of these three terms [hormone, cytokine and protein hormone] mean and how are they different?
Both cytokines and hormones are a class of signalling molecules that are secreted by cells. The difference is that cytokines act locally, are produced by many cell types and typically have a function in the immune response. Hormones are secreted into the blood by dedicated endocrine cells so the can regulate other cells across the body.
Do any of the three terms represent a superset of the others?
Protein hormones are a subset of hormones. An other important subset are steroid hormones.
It can be argued that cytokines function as hormones in some situations. For example: in the case of an infection high concentrations of cytokines can regulate body temperature causing fever. wikipedia
Are the terms context specific at all?
In some situations it can be argued that cytokines act as hormones. But the feature that groups them is that the are an actor in the immune response.
Classical hormones are produced by a gland and excreted in the blood. This is not context depended.
Are cytokines actually fundamentally different from hormones?
All Cytokines are proteins and don't act fundamentally different from protein hormones. Both have to act through membrane bound receptors since proteins can not pass the cell membrane on their own. Steroid Hormones can pass the membrane and typically have intracellular receptors.
Are there any canonical examples of each?
Interferons are a large class of cytokines. IFN-γ is a well known example. Androgens are steroid Hormones . Testosterone is one of them.