> It seems like
>
> (setq add-log-current-defun-function nil)
>
> should suffice.
That does not stop add-log-current-defun for working, in absence of an a-l-c-d-function it just uses the regexp, which is what is returning the spurious name.
Either adding a custom add-log-current-defun-function, or setting add-log-current-defun-header-regexp to a buffer-local value so it recognizes filenames would work. (Buffer-locally, because the variable is in fact customizable so we can't be sure what its global value will be.)
This works, for example:
(defun xref--add-log-current-defun ()
(if-let (item (xref--item-at-point))
(xref-file-location-file (xref-match-item-location item))
(xref--imenu-extract-index-name)))
(setq-local add-log-current-defun-function #'xref--add-log-current-defun)
> Is that the intended behavior, to report the absence of the current defun?
The final intended behavior, IIUC, is that the filenames in the xref--xref-buffer-mode get passed to which-function as "defun" names.