Inner and equi joins are not self-joins per se. A self-join joins the table to itself. This is always a form of INNER join.
the term inner join has been coined to differentiate between inner joins and OUTER joins. It's true that an equi-join uses in the join-predicate. You should prove to yourselves that not equal (<> or ! ) in a join will give you a cartesian product in the result.
An equi-join can be either an inner join or an outer join. The true definition of equi-join (in addition to the relation in the predicate) is that both join columns appear in the result. If one is projected out the equi-join is called a natural join. If other operators are used (< > which is rare in business databases) the join is called a theata join. both the equi join and the natural join are forms of the more general theata join.