Akeeba Backup displays the message in response to the version PHP itself reports. We don't do any "discovery". We literally just read the PHP_VERSION constant which is hardcoded into PHP at the time it was compiled. This means that the version we report is with absolute precision the version you are using on your site.
That said, I do not want you to believe me blindly. I want you to see for yourself.
Go to your site's backend. Click on the menu item System, System Information. Now click on the PHP Information tab. What you see there is the output of PHP's built-in phpinfo() function. It tells you the version of PHP you are running in big type at the top, next to or right underneath the PHP logo (depending on your PHP version).
Now that you see that you are indeed using PHP 7.3 go back to your host and ask them how to upgrade the PHP version to 7.4.
If they tell you to use a tool in their hosting control panel (which they seem to have implied in their response you quoted) do remember that their tool modifies your .htaccess file. If you are using Admin Tools' .htaccess Maker you will need to transcribe these lines into Components, Admin Tools, .htaccess Maker, Custom .htaccess rules at the bottom of the file. Typically, these lines look something like this:
AddHandler SomethingHostSpecificHere .php .php5 .php7 .inc
or something like this:
If you are unsure what the relevant lines are, please ask your host.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!