There are domains that are not norm-Euclidean but were proven to be Euclidean using Motzkin's transfinite Euclidean function. In this answer (let's call this answer 1), https://math.stackexchange.com/a/2154963/364725, they used the transfinite Euclidean function $f$ and said that $f(r)$ can be computed given an element $r$.
However, from this answer (let's call this answer 2), https://mathoverflow.net/a/255519/196685, they said that among the non-norm Euclidean real quadratic number fields, only $\mathbb{Q}(\sqrt{69})$ has a known Euclidean function. Does that mean that the transfinite Euclidean function $f$ is not considered as a Euclidean function? I'm confused because, in answer 1, it looks like they were saying that we can perform the Euclidean algorithm there using $f$ (because they said $f(r)$ can be computed)?
In sum, what I want to know is, what are the Euclidean domains where we can actually perform the Euclidean algorithm? (Also, I'm not pertaining to just the quadratic number fields. Even a discussion for higher degree number fields would be great.)