I would say it's IDB. Note the canceled flight earlier in the day. Had there been seats available, why wouldn't they have flown OP and family? Therefore, I conclude the flight was entirely full, hence oversold.

The fact that the airline arbitrarily changed OP's tickets is irrelevant. They did so in order to have more seats available because they needed those seats, and assigned them to others. Therefore, oversold and IDB.

I'd file a complaint with the Department of Transportation, which accomplishes two things. First, if DoT agrees, it's IDB and compensation will be legally required. Second, it counts as a complaint against the airline in DoT's statistics. (If I were sufficiently annoyed, I'd file a separate complaint for each passenger.)