As I see it, (2) has two possible readings. "That day" in (2) could be substituting for the use of "today" in the reported sentence. On this reading, "that day" modifies "weather":
direct: She asked me, "How is the weather today?" reported: She asked me how the weather that day was.
Alternatively, (2) could be equivalent to the following, in which "backshift" is at work. We may suppose that she asked how the weather on some previous day ("that day") had been.
direct: She asked me, "How was the weather that day?" reported: She asked me how the weather that day had been.
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