Last note on vests: As vests evolve from a semi-undergarment to one that is worn as an outerlayer, it's styles change as well.
As a result, there are now many more choices for the the backing than the binary choice of lining/facing.
For example, I have a vest from Jon Ashe
that I bought last year.
The front is in a washed brown wool herringbone. The lining of the front is in purple cupro.
The lining of the back is brown and white striped shirting. The back is brown cotton moleskin.
So each of the four areas was handled differently, but good design produced a happy combination.