Best-practice for (not?) re-using/caching computed --with options

Commonly (?) --with-foo is used in the following way:

--without-foo: disable foo, don't do any checks for it
--with-foo=bar check for foo using bar, if it doesn't work, error
--with-foo=baz check for foo using baz, if it doesn't work, error
--with-foo=check: check if foo works with bar or baz,
                if it doesn't work, disable
not specified: same as with-foo=check
--with-foo:    same as specified, but if it doesn't work, error


This matches the documentation for AC_ARG_WITH
https://www.gnu.org/savannah-checkouts/gnu/autoconf/manual/autoconf-2.71/autoconf.html#index-AC_005fARG_005fWITH-1

All of --with-foo, --with-foo=bar and --without-foo are re-used (when 
configure is re-run by make, then those will be used, again) and the 
test for them may _should_ be cached (I _guess_ this is a best practive 
but I did not found that in the documentation).
Please correct me if anything so far is wrong.

But when re-run by make without being specified on the original 
configure command line, then the check for foo would be re-run, which 
could lead to the program being configured different then before (for 
example because a system header was added/removed).

I do search for a "best practice" solution - should the check in 
configure for foo cache its result as-if --with-for / --without-foo was 
specified by the user?

Would it be more reasonable to not do any caching if --with-foo=check is 
implied, so the check is always done?
Should this be different if --with-foo=check was explicit specified 
(which I _guess_ could be test with some _set variable)?

Is there an option to "add" to the configure options that are used for 
re-running the configure script?

Thanks for any pointers,
Simon