How are the two definitions of graph state mathematically equivalent?

The first definition says the nodes start in the $|+rangle$ state and then you apply a CZ for each edge.

The $|+rangle$ state's stabilizer is $X$. When you conjugate an $X$ observable by a CZ operation, you end up with $X otimes Z$. When you conjugate a $Z$ observable by a CZ you just get the $Z$ unchanged. This means propagating the $X$ stabilizer forward through time, through the CZs, kicks out a $Z$ on the other side of each CZ. So you end up with a stabilizer that's $X$ on the node times $Z$ on each neighbor. That's the second definition.