They do use cPanel... and you, sir, are my hero for pointing me in the right direction!
The ModSecurity section on my end has a list of all the subdomains my site uses, and I can switch ModSecurity on/off for each individual (sub)domain. Oddly enough, the site and all its subdomains had this feature turned "off" except for the new moodle install I was trying, and I haven't ever used this part of cPanel until now, so I haven't turned anything on or off.
I switched it off for that install subdomain and the warning on the moodle install disappeared!
There wasn't an option for me to do anything about the server signature, but I'll contact them and see if they can, and reply again with how it goes -- for future reference of those who may have similar issues and find this post.