Why do the PHP versions shown in LiteSpeed's "External Applications" not show all PHP versions installed on the server?
On a cPanel server, LiteSpeed will automatically detect the PHP handler and PHP version installed in the cPanel MultiPHP Manager/CloudLinux PHP Selector. There is no additional external application or script handler required unless you have a requirement to override settings.
New installations of LiteSpeed will not have any external apps. If upgrading from earlier versions, some external apps and script handler settings may be carried over. Sometimes certain lsphp versions, such as lsphp73 or lsphp74, may be missing. This is absolutely fine.
More information on this can be found on from LiteSpeed's Documentation, here.