I can login to a root shell on my machine (yes or no, or I don’t know): yes
I’m using a control panel to manage my site (no, or provide the name and version of the control panel): yes, 1und1 for VPS and Plesk for hosting my domains
is there such a thing like a free security certificate from Let’s Encrypt, that is trusted by standard browser Chrome, IE, FireFox, respectively Windows or OS ?
It’s about avoiding a recurring certificate fee for signing up to a service where any good or bad can sign to anyways. I mean in terms of trustfulness of the domain.
Is it possible to avoid fees and domain visitors won’t see the warning ?
The certificate you’re serving on port 8443 was not issued by Let’s Encrypt - it’s self-signed. This appears to be a Plesk installation, and I imagine the steps you need to go through to use a certificate from Let’s Encrypt for the control panel itself are separate from the ones you have to go through to use it for a site on your Plesk installation (where you’re indeed serving a valid certificate from Let’s Encrypt).
I don’t personally use Plesk, so I don’t know exactly how you would do this, but perhaps this’ll point you in the right direction.