First, elipsis is ellipsis. Second, where is one example of ellipsis?
Quote:
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1 -We have always thrown a party on his birthday, except for two years ago, and that was because he was away on a business trip.
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What is the referent of
that? The fact that we have always..? Or the exception? Your sentence is ambigious to me.
We have always thrown a party on his birthday except for two years ago
when he was away on a business trip.
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Quote:
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2-We have always thrown a party on his birthday, except for two years ago, because he was away on a business trip.
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No good.

Again, what does 'because' relate to? What does it explain?
My reading of your second sentence is this:
We have always thrown a party on his birthday because he was away on a business trip, (except for two years ago.) -- Highly unlikely.