Tea

I would still call it flavored tea. If the tea only contains dried pieces of plants, without added extracts, etc. it falls in the category of traditionally flavored teas, in my simplified categorization. Those pieces can also be called inclusions.

Ginger tea is something you could make at home. Buy some tea leaves and either fresh or dried ginger, and mix them together. Jasmine tea is more difficult, as dried jasmine flowers loses their taste. To make jasmine tea, you either need to use extracts or you need to use fresh flowers which are allowed to dry while mixed with tea leaves. Tea leaves are like sponges: they absorb scents/odors. In many commercial teas, jasmine flowers are used only for visual appeal. The taste is added with extracts.