Mathworld.wolfram.com claims that the definition is "for any closed and connected A,B such that A cup B = X, the set A cap B is connected." Since I have not seen the definition anywhere else, I did not dare correct Wikipedia.
There seem to be two problems (1) the lack of the requirement connected for A and B. (2) Calling A,B a partition even if A cap B may not be empty.
The above are both valid criticisms. Point (1) is valid because without this requirement the definition doesn't actually work, as almost no sets satisfy the weaker (incorrect) definition. Point (2) is valid because of the definition of partition, which specifically requires that the subsets in question be disjoint.
Using Mathworld as an authoritative source on this topic seems best, so I'm going to change the page appropriately.