You'll find your answer and much more in the little-known paper  which considers in-depth a surveys all of the dozen different known ways of axiomatizing Euclidean rings (including those of Nagata and Samuel), and explores in-depth all of their logical interrelations. It's a useful convenient reference to have at hand when you're comparing texts which use (seemingly) different definitions of Euclidean rings / domains.
 Euclidean Rings. A. G. Agargun, C. R. Fletcher
Tr. J. of Mathematics, 19, 1995, 291 - 299.