The next possible reason is that your licence has expired. If you have LiteSpeed web server running and there is no valid license every request will result in a 403 or LiteSpeed web server is switching to Apache.
Again, request method has nothing to do with "CURLOPT_NOBODY". This curl option has affect on returning message body and depending on the value for this option the ouput of the request will be returned or not. For caching a page you don't need the return and that's why it is better to set value...
This is not a HEAD request. HEAD means only HTTP Header and no message body, but message body is not HTML body. That is a big difference. A only HEAD request prevents to cache a page.
You seem to talk about cache plugin settings of Wordpress? If so, you should know that these forum doesn't support cache plugins, even they are made by LiteSpeed itself. To get support for your issue please check this...
Maybe I am wrong, but I got an email notification from Google user group that a new version has been released, but I just checked it. I am wrong. This notification is about a user who writes a post about version 1.4, so sorry! :(
Of course it is matters for both. Unsufficient cache rules causes wrong cached pages. I checked your page today and found some wrongly defined cache rules regarding apple devices.
No, I use completely different data. User Agent is yesterday, I use "client hints"
https://web.dev/i18n/en/user-agent-client-hints/
https://developer.mozilla.org/en-US/docs/Web/API/User-Agent_Client_Hints_API
https://developer.mozilla.org/en-US/docs/Web/HTTP/Client_hints
Yes, for Multithreading Magic Crawler. :) I don't really need these cookies, but they make it easier to define safe cache rules. The current are not really good.
Another question: If I check your page and depending on used device there is a _lscache_vary cookie set. Where does this cookie come from? LScache extension for OC?