I agree, it's more complex because there's at least a few different criteria that all combines together:
- Who owns the IP
- Who's controlling the direction of the game
- Who's doing the work
- Who's paying the bill
For instance with Perfect Dark, Microsoft owns the IP, The Initiative controls the direction of the game, The Initiative and Crystal Dynamics are doing the work, and Microsoft is paying the bill.
The answers are similar with project Dragon Dragon and IOI Interactive but Xbox Game Studios instead of The Initiative. (I think that's the project codename).