Another "that" question

This question represents the opposite of the "that that" topic. I am wondering about the disappearance of "that" in certain clauses.

For example:
He said I had to meet him later

As opposed to

He said that I had to meet him later.

(Better examples abound, but I think [that] you get the idea)

I was always under the impression [that] the second example was correct, or at least, more formal. My Latin professor always obliged us to use that in our translations, even though colloquial usage said otherwise.

Is this variability a question of style--standard versus formal--or is the second typology obsolete?