The difference that is confusing people is the difference between procedural equality and substantive equality (or 'outcome' equality). The picture posted above from the teacher I don't find helpful, because it implies that there isn't such a thing as substantive equality.
Basically, procedural equality is about giving everyone the same thing, regardless of need or circumstance. Substantive equality (sometimes called equity) is about trying to achieve equality of outcome - so people might be treated differently based on need or circumstance. They're both types of equality; one is just more meaningful in real life than the other. Our legal system tends to focus a bit more on substantive equality than procedural equality.
In my textbook (printed by CPAP) I define both of them and explain the difference. It's really not a contradiction: it's just applying 'equality' to different stages of the process.
Also, with the difference between equality and fairness, it's really just that equality is a small part of fairness. Fairness requires some measure of equality (usually substantive equality), but it also requires other things. Fairness is a measure against values and what people think 'appropriate' treatment and natural justice demand; it includes equality, but also dignity, impartiality, access, etc.