[Back In The Saddle]

Jurassic alludes to another important point: The clock being started/stopped and the ball being alive/dead are entirely independent.

You said "...occurring before the clock is started following the first..."

There can be a lot of basketball played, and a lot of non-basketball things happen, before the clock starts again. As JR mentioned, you could have free throws, during which the ball is live, and you could have a common personal foul committed. Or you could have a throw-in, and again, a common foul committed. Or even possibly a jump ball following a the first foul, during which a common foul may be committed.

And I totally agree about the terminology. You need to know what a false double foul is, but you'll never call or report a "false double foul." You just need to know it so that you aren't confused when it happens and try to apply the double foul penalty. Penalize them in the order they happened, that's the part you really need to internalize.