People get lazy about writing, and might not use words scientifically. But if the farmers are all using a phrase in one way, and the scientists are using a phrase in another way, who is right 1000 farmers, or 10 scientists?
The definition that I use as a farmer when I say "winter wheat" describes planting a crop in the fall, and it goes through winter as young plants and I harvest it in early summer. I often think of it as growing under the snow. In many cases, I could plant the same wheat in early spring, and it would go through its life cycle a little later in the season, and I'd harvest a slightly less productive crop in late summer. I might be able to grow the fall-planted wheat without
irrigation, but would have to water the spring-planted wheat. I often grow the same variety both as a winter wheat and as a spring wheat.
There are other wheats which a scientist would call "winter wheat" which require so many hours or degrees of chilling before they will set seeds. They won't make seeds until they have gone through a winter.
When I plant wheat within a few days of the snow melting, they still
experience 2.5 months of frosts, so maybe that is
enough to make them believe they have gone through a winter.
If you have a bag of wheat, then there is plenty for planting both in the fall, and in the spring.