Here are two dialogues. What would be the difference between あったんだった and あったんだ?
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This use of た is called modal-ta or 発見のタ. There are a number of questions related to this on this site.
"fixing (or updating) one's previous knowledge/expectation" seems to be the key
It is used when one's expectation has just matched or failed to match reality.
In Dialog 1, B is updating his own knowledge, and that's why the sentence ends with the modal-ta. But 会議があったんだった sounds a little redundant to me. 会議があるんだった is equally correct and probably more common.
In Dialog 2, B is trying to update A's knowledge by telling what is already known to B, so the sentence must not end with the modal-ta. 別の予定があるんだ and 別の予定があったんだ are equally correct, but the latter contains a modal-ta before the end of the sentence to update A's knowledge; it implies "(I know) this is not what you've expected".
In such cases, there are two relevant differences to consider:
In the first scenario, in which two co-workers talk about a conference, B already knew that a conference was to take place on that particular day. They simply seem to have forgotten and are now re-establishing this knowledge. In that case, the original information about the conference came from A and is now re-stated by B.
In the second scenario, the information that B already has plans is new information to A. Therefore, it is in the present tense, as B is offering this new piece of information to A just now.