To confuse you even more here is a third definition given by R L Moore in c1920.

Two nonempty sets are separated if neither one contains a point nor a limit point of the other.

(Of course, Prof Moore did not say nonempty- he did not believe empty point sets existed.)

A set is connected if and only if it is not the union of two separated sets.

But here is the kicker: All three are equivalent. You ought to prove it.