All of the types of equality (or inequality) you've listed are
politically important; and I agree that we have to distinguish
them and further analyze each of them.
The word "equality" is subtly ambiguous. Think of equality
of taxation. What is it? Does it mean that everyone pays the same number
of dollars in taxes? Or does it mean that they pay the same percent
of their income as tax? Or that everyone bears an equally heavy
burden (which might mean a higher percent for people with higher
incomes)? To conservatives equal taxation means one thing, to
liberals another. So what is equality?
Each political perspective has its own definition of the politically
important kinds of equality. Classical and modern liberalism,
for example, have their own views of each kind of equality
that John mentioned; and I tend to disagree. Concepts like equality
and freedom have been called "essentially contested concepts."
They are so important philosophically and politically that we
constantly debate their contents.
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