I've finally moved the site over to the https protocol, and I didn't have to spend a single euro!
I used the linux foundation project letsencrypt.org, which through a two-way check lets you get a 90-day certificate that renews itself (automatically!) every 60.
All of this works through a check performed at issuance time. Every domain name you want to certify (they can also be multiple domains, SAN etc) is verified through some files created during issuance.
The program to use for issuance is called acme.sh, full guide here: acme.sh
Basically, when you request a certificate from letsencrypt.org, for example for nutellino.it ๐ , you also need to specify the site root where the check files will be created automatically.
This way letsencrypt will connect to the website http://nutellino.it and check, inside some folders created ad hoc, whether the files are there.
So if the files are there, it means whoever is requesting the certificate for a domain is definitely either its owner or its administrator.
So if we wanted to use acme.sh to issue the certificate for the nutellino.it site, we'd just run this command (once acme.sh is installed):