The name "Jasmine" basically refers to some of the many varieties of shrubs (above), vines (like sambac, pink and confederate) and small trees (like Lakeview). They are all blessed with fragrant white flowers many people and insects enjoy. True jasmines are related to the olive, privet and osmanthus.
Many plants called jasmine are not jasmine but closely resemble the classic flower appearance and fragrance. In the list at the bottom you will see some of the many plant types (Latin names) said to be a jasmine.
I would post pictures, but I'm all soothed and sleepy from dinner. Night-night.