I'm guessing it's a matter of direct vs. indirect?
My older brother said "Just a minute, please"
My older brother said just a minute.
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Both って and と are quotative particles and are grammatically interchangeable. While the former is colloquial and casual, the latter is not particularly formal nor casual (and it's thus "safer" while you're a learner). In your first example, 言ってた is omitted because it can be inferred from the context. This omission can happen regardless of whether it's preceded by って or と.
Therefore, the following sentences are all valid and equally natural: