Euclidean function of Euclidean domain defined at 0

You'll find your answer and much more in the little-known paper [1] which surveys all of the dozen known ways of axiomatizing Euclidean rings (including those of Nagata and Samuel), and explores in-depth all of their logical interrelations. It's a convenient reference to have at hand when you're comparing texts which use (seemingly) different definitions of Euclidean rings / domains.

[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

I think this is only because you don't care dividing by zero. But you may as well define the value at zero to be 0, and any other value at least 1. See this expository paper by Keith Conrad for interesting remarks on euclidean domains and functions.