> Does it work to test whether server-process is bound and non-nil?

It works, indeed.  Would you accept a documentation PR on the variable
and the two functions, which are much more obvious candidates?
Something like:

For a general way of determining if the current instance has a server
running, check the value of `server-process`.

Or a PR adding a stupid helper function like:

;; server.el ---
...
;;;###autoload
(defun server-started-p ()
    "Return non-nil if this Emacs has a server started."
    (and (boundp server-process) server-process))

What do you think?

Th