"That" signals reported speech. No "that" without a comma also signals reported speech. No "that" and a comma expects a direct quote.
He said that we should meet.
He said we should meet.
He said, "We should meet."
Style guides stress brevity: The shorter the better. "That" is often a filler word. Using this criterion, #2 is better than #1. Sometimes "...
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.
Ned: How old were you when you decided to become an astronaut?
Mr. Nash: I was 12 years old and I was in a physics class.
Indirect or Reported Speech:
Ned asked Nash how old he (Nash) had been when he had decided to become an astronaut.
Nash replied that he had been 12 years, and that he had been in a physics class.
The reported verbs are ...