The effect caused on tannic woods like Oak with the Rust/Vinegar mixture is a chemical reaction causing a staining effect, it is not a dye or ink. Though Oak Gaul Ink is the same stuff.
An Iron Nail hammered into an oak tree will cause the same effect as it rusts, and as Pete says from his rusty water in a skip, the longer it stays there, the more the effect spreads.
The depth of the reaction in the wood of a bowl is not that great, as the application of the ebonising liquid (I've always known as Liquid Nightmare) is not that prolonged. As to whether it's food safe, I cannot comment, but on the outside of a bowl as you suggest I cannot imagine it being a problem.