Hans, unfortunately that does not work: it just makes \Arrowvert act like \| (\Vert).
The TeXBook explicitly documents why these have "arrow" in their name, they are the vertical bits of the vertical arrows, without the arrow heads, and it also documents that they are wider than the regular delimiter.
TeXBook says:
The control sequences ^|\arrowvert|, ^|\Arrowvert|, and ^|\bracevert| produce
delimiters made from the repeatable parts of the vertical arrows, double
vertical arrows, and large braces, respectively, without the arrowheads
or the curly parts of the braces. They produce results similar to
^|\vert| or ^|\Vert|, but they are surrounded by more white space and
they have a different weight.
Compare the two versions above the vrule using pdftex with this modified version of your example.
\nopagenumbers
$\| \Big\| \Bigg\| \big\| \bigg\|$
$\Arrowvert \Big\Arrowvert \Bigg\Arrowvert \big\Arrowvert \bigg\Arrowvert$
\hrule
\delcode`\| = "26B30D
$\| \Big\| \Bigg\| \big\| \bigg\|$
\def\arrowvert{\delimiter"26B30C }
\def\Arrowvert{\delimiter"26B30D }
$\Arrowvert \Big\Arrowvert \Bigg\Arrowvert \big\Arrowvert \bigg\Arrowvert$
$ \mathchar"30C \mathchar"30D \mathchar"33C \mathchar"33D $
\end
when the delimiter goes to the extension font the requested size is already larger than the glyph in the specified slot so classic tex uses the repetition formula which unusually only has the repeater and doesn't use the specified slot at all, but luatex is using the specified brace part which (I think) is wrong but haven't traced down a clear tfm specification to be certain.
David