You need to remember here that DL does not have any "First Class" on international routes. They have "business class" which equates to Delta One on DL operated flights and Prestige Class on KE operated flights. If you want true F, you need to purchase via KE directly and fly on a KE marketed/operated ticket on a route and aircraft that KE offers F. Purchasing a ticket on DL7851 is really just buying a J ticket on delta, which equates to Prestige on Korean.

I agree that to someone not well-versed in the specifics of what each airline offers, this can be misleading. But it's unfortunately correct.