So if David had chosen the U.S. side...

No, because there were no real missiles being launched.

The question really is - would Moscow have detected an attack, and then launched theirs?

I think only Joshua's moves would actually happen. Since Joshua didn't have access to Soviet systems, the scenario may have caused some confusion but ultimately no damage would be done.

I'd say no, only because, as McKitrick states, the silos won't accept the launch codes unless they are at DEFCON 1. In other words, he would have tried to launch the missiles, but it would only work if he could convince the General to go on high alert. That's why it was so important that the simulated Soviet launch go first, to trick them into going to DEFCON 1, at which point he could lock them out of the system. (We could ask why he doesn't have the ability to change the DEFCON status, but I guess we just have to accept that it's a built-in safety check).

If things played out the same and WOPR obtained the U.S. Launch codes then the Soviets would have detected the launches and fired back. In fact in this scenario the U.S. probably isn't even aware it's happening so they wouldn't have any assets to stop it.