I'm not knowledgeable but with limited experience. Perhaps the small ginger tubers are sold with foliage as the foliage can also be used in dishes. It is aromatic as well.
I have planted store bought ginger which has never bloomed for me. I understand if it does, they are small, insignificant blooms.
I think these photos represent different types of ginger but get lumped into a generic ginger category in the Plant files. The last bloom photo is a shampoo ginger which I don't believe is edible.
If your starts are putting up foliage, I would plant them horizontally about one or two inches deep. They will develop more roots which you should be able to leave in soil all winter. Then you can dig up palms of ginger as needed for cooking.
