Till some extent you are right, Engagement band is Ring with Design on all the side of the ring which means even if you rotate a band on your fingure You will still see the same design, it is not necessary Engagement band should be plain, I may or may not have Diamonds or gemstones mounted on it.

Now in case of Engagement rings , they only have designs on one side of the ring so if you rotate the ring around your fingure the design will be covered in side your hand towards your palm. Again an engagement Ring may or may not have diamonds and gemstones mounted on it.