In backend of Moodle there is a field to change the Login URL, but I found no field to change the standard Signup url (login/signup.php). I've muddled through the core code and changed in several files temporarily the standard URL into a specific URL. But sure this is not a fine solution. Is there a smart way to do this with e.g. an override, or with a small auth-plugin?
You can not change the URL using standard methods available in moodle. What you can change is behavior and presentation.
Signup form is implemented in lib/authlib.php in method signup_form(). To override this you would need to create new auth plugin that overrides this method and enables user signup. See auth_email for example of implementing user signup.