This page states that I marked the question as a duplicate, but I never did:
Unlike this question, I don't think all the different reasons should be listed. I just don't want the statement to be false.
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The close reason displayed is the one chosen by the majority of voters. So even if you chose a different reason then it would still display "closed as a duplicate".
The only exceptions to this are when someone uses the gold badge duplicate hammer on a question or a moderator votes to close. Then that overrides all the other close votes regardless of how many there are.
There is a difference when more than one duplicate question is suggested. In this case it displays all of the possible duplicates in the header.