The $n$th prime number is the smallest difference of products of distinct previous primes. For $n\geq 2$, the smallest positive prime number formed by multiplying $2, 3, \dots, p_n$ and up to $1$ subtraction / addition
$$
2 + 3 = 5 \\
2\cdot 5 - 3 = 7 \\
3 \cdot 7 - 2\cdot 5 = 11 \\
5\cdot 11 - 2\cdot 3 \cdot 7 = 13 \\
7 \cdot 13 \cdot 2 - 3 \cdot5 \cdot 11 = 17\\
\vdots
$$
is always $p_{n+1}$.  Has this been proven?

You stopped your examples one step too early. The smallest number $>1$ obtainable from $2,3,\ldots,17$ is 
$$2\cdot 5\cdot 7\cdot 11-3\cdot 13\cdot 17=107.$$
(I.e., $663$ and $770$ are the factors of $N:=2\cdot 3\cdot 5\cdot \ldots\cdot 17$ that are closest to $\sqrt N$).
