The environment check in Moodle checks only the Moodle code/DB - the DB named in config.php of moodle code. It will not check WordPresses!
Uhhh .... gave you a way out ... and you took it! :\
Most of those commands really don't need to be obscured! But they are absolutely necessary for you to know if ... if ... you are going to attempt to follow moodle docs where you see command line.
Case in point ... uname -an will tell us what linux distro your shared server is running. Makes a difference ... if server is a CentOS based Linux, you might not be using 'sudo' in front of all those commands. And to install software, which you might be able to do, one won't be using apt-get! Plus, and this could be a shocker ... you might be running a distro which is near EOL!!!! So your future plans to build or lease might be sooner than you planned.
Another case in point - cron job. That's setup outside of Moodle and does require you to know the path to php-cli. php -v would have verified/shown that path to be valid.
It's also important to know that path is accurate to php-cli when executing the scripts in code/admin/cli/ to fix your database character set/collations issue. Tried that yet? Please do!!!!
php -m shows the php extensions installed.
What not to share ... maybe the url to your server, although that could be discovered via Google easily enough! (the 'blackhats' probably already know and have been poking and probing your WordPresse's for a long time! BTW, a vulnerable WordPress can be the cause of a hacked Moodle!) Obviously, any command that shows login/pass ... like a mysqldump command.
Sorry ... but the comment about 'if it ain't broke' ... even though this entire thread started with site being broken and we are just now discovering where and how bad ... that philosophy ... 'if it ain't broke' is one of the things that will lead to a site that is vulnerable and, sorry to say, eventually 'hacked' - and many times, the OP/Admin is blissfully unaware. Goal of hackers/crackers now-a-days is not to disclose they have hacked your server. They have gone beyond defacing the front page, but rather, will steal any user data you have or try to use your server as part of their bot network.
Ok, will get off my 'soap box' now!
'SoS', Ken