I remember being confused by this when learning to fly.
I was taught your 'other side' which accounts for inertia, though it was not described in these terms, which might have helped my understanding of what I was being told. I couldn't see it, arguing the 'moving parcel of air' view and I recall my argument used the example of circling a hot air balloon.
When I eventually figured out by myself what the instructor was trying to convey I did at least acknowledge the possibility but maybe I've never made a sufficiently abrupt turn down wind but I cannot say I have ever noticed the effect of inertia in that situation.