See the paper [1] which considers in-depth a dozen different ways of axiomatizing Euclidean rings and explores all their logical interrelations. It's a useful reference to have at hand when you're comparing texts which use (seemingly) different definitions.

[1] Euclidean Rings. A. G. Agargun, C. R. Fletcher  
Tr. J. of Mathematics, 19, 1995, 291 - 299.  
http://journals.tubitak.gov.tr/math/issues/mat-95-19-3/pp-291-299.pdf