If Delta promised you a flight voucher, then it should have given it to you. But it’s not entirely clear to me what the Delta agent was trying to do when you were rebooked.
At any rate, the time to settle that question would have been before you left, and not afterward. Any airline is going to be reluctant to offer you a voucher based on a kiosk interaction. That would have to be verified by a gate agent and settled before your departure.
It’s unclear what happened before your Detroit-bound flight departed. Were they trying to prevent a delay (if they were, they actually made things worse) or were they trying to remove two passengers from an oversold flight (if that’s the case, then they misrepresented the facts when you called).
I’m fairly certain that some airlines fudge the circumstances of a delay to suit their interests. For example, a broken air conditioning – a “mechanical” delay – can be turned into a weather delay, because it lets an airline off the hook for covering rooms and meal vouchers for displaced passengers. But I don’t think that’s what happened here. I think this was just a misunderstanding.
I contacted Delta on your behalf and after reviewing its files, it offered you two $400 flight vouchers.