Description of the need When anonymous users are set to post comments, then additional configuration settings for anonymous commenting appear on the content type settings. Two of them allow anonymous authors to leave links to their homepages.
In most cases the field is abused by spammers who put ad links in that field. Despite it is possible to easily hide or remove the field using template overrides, I believe this field should be properly changed in core.
Proposed solution I would completely remove the field, however considering those rare websites which need visitors to leave their homepage links, we could make this configurable just in the same way as the comment titles are allowed or not like shown on the screenshot for proposed change.

Recent comments
I'll test this tomorrow to see if I can replicate the problem and report back. It's not "per user" so it should be working.
TFA - how to setup fallback options 'per user'
As I work through things I flush the caches after every change I make. As far as I can tell I have the custom module in the correct place (/modules/custom/fix_zero_search) and the file...
custom module not showing up in modules list
If you haven't already, flush all the caches. If it is in the modules folder whether directly or in a custom folder it will show up. If not, something else is wrong with your site; check...
custom module not showing up in modules list