I always just took it as two different names that got tossed about.
It seems like most manufacturers used the term "full metal jacket" except Remington, who has used "Metal Case" for decades to describe about any bullet with no exposed lead.
(Well- at least no VISIBLE exposed lead when loaded in the case. Any of Remington's bullets I know of which they called "Metal Case" or "MC" had exposed lead bases.)
I think the term "full metal jacket" implies there is no exposed lead anywhere on the projectile, but that's obviously not how it is. I think the name would be more accurate if they removed the "full" from the term.
Since I'm being nitpicky, I suppose Remington's "metal case" is a better term since the bullet is encased in metal, even if it isn't completely encased.
Ah, the things I take time to think about on a Sunday when my football team isn't playing.