contrary
in
Waiting for Godot

different (perhaps opposite or mutually exclusive)

In formal logic, contrary propositions or contrary arguments describe two things that cannot both be true. They might be described as mutually exclusive. For example, a pet cannot be both a cat and a dog. It doesn't need to be either, but if it is one, it cannot be the other. So the statement that a pet is a cat and the statement that the same pet is a dog, are contrary statements.

disagreeable -- typically in reference to someone's personality

in opposition to

You could also think of this as meaning, despite, or in spite of, or in conflict with; but since a common meaning of contrary is opposite, you many find it easiest to think of contrary to as in opposition to.

an expression used to intensify denial of an idea

More rarely, on the contrary is used to oppose an idea contained in what was just said rather than to oppose what was literally said. Here is an example:  "I will not pay you for the work. On the contrary, I may sue you for damages."  Suing for damages does not oppose not paying, but it does oppose the idea of paying.

with an opposite or different effect; or something with an opposite or different effect