If you look in the literature, there are multiple different definitions of Euclidean domain. A lot of the differences has to do with how a Euclidean norm is defined at zero. People seem unnecessarily averse to the idea that zero should have largest norm (corresponding to the fact that in the reversed lattice of ideals in a ring, the zero ideal is at the top), and thus the image of a Euclidean norm shouldn't be restricted to $\mathbb{Z}_{\geq 0}$.

In Lenstra's unpublished Lectures on Euclidean Rings, if this type of ordering is used, then the "minimal norm" on a Euclidean domain is shown to be logarithmically superadditive (see Corollary 2.5). Many other natural properties and results flow from this more natural definition.