What exactly is being talked about here?
"g(x) = the largest power of 2 dividing x".
When x is 12, we get that g(x) = 4. The largest power of 2 to divide 12 is not 4 ($16$) it would be $2^2$ or 4. 8 does not divide 12, neither does 16.
What am I misunderstanding?
$g(x)$ is equal to the actual power of 2, not the exponent. So $4=2^{2}$, which is the largest power of two dividing $12$.