What is the difference between a cytokine, a hormone and a protein hormone?
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. 

Cytokines are a group of small protein that  have a fundamental role in the immune system. They are typically short lived and typically have local effects and are specific cytokines are often produced by multiple cell types. 

For example they cause local cells to generate heat and cause blood vessels to become wider and more permeable to blood cells. parham (2015) Some cytokines can act over longer distances, IL-6 for example cause local inflammation and regulates fever by targeting the  hypothalamus causing it to increase body temperature overall. For more information check this link.

Hormones are secreted into the blood by dedicated endocrine cells so the can regulate other cells across the body.(alberts, 2008) 

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. IL-1α, IL-1β and IL-6 are good examples here

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. This wiki discusses some differences.

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 and Interleukins are a large classes of cytokines. IFN-γ and IL-6 a well known examples.
Androgens are steroid Hormones . Testosterone is one of them.