One can write a sentence with a partial quotation:
Direct speech: We live in a madhouse! We have to move.
She says they "live in a madhouse" most of the time.
This is unobtrusive: the quotation is direct speech, because the exact words are repeated, made to fit into the syntax of the sentence, which was easy enough. It is a predicate, so it fits ...
(1) I agree that the sentence you mark as wrong is much better without the that; (2) do not look to Wikipedia for good English. I would leave the that out in both sentences, the one you marked wrong and the one you quote from Wikipedia.