Euclidean not implies norm-Euclidean

Statement
A ring of integers in a number field may be a Euclidean domain, even though it is not a norm-Euclidean domain. In other words, it may have a Euclidean norm which differs from its norm function, even if it is not Euclidean under its norm function.

Example
The ring:

$$\mathbb{Z}\left[\frac{1 + \sqrt{69}}{2}\right]$$

is a Euclidean domain, but is not norm-Euclidean.