Seems to me that if there are two ways to do things, you had best learn both. When I hear the we do it this way and they do it that way discussions, I just laugh. And guess what, I'd say there are more than two ways to do koshinage, or in fact, any technique.
I just watched koshi waza videos from Kyuzo Mifune who they called God of Judo. Seems he does koshi techniques with one foot!