I’m still having trouble following your question.
You say you’re familiar with giving WP it’s own directory, but I think you may be confusing installing a second WordPress install in a separate sub-directory with giving WordPress it’s own sub-directory.
Possible file structures:
Single site:
/index.php (and all the rest of wordpress installed at root)
Multiple sites on a server:
/index.php
/wp-????? (a complete wordpress site, all your wp-???.php files, wp-content, wp-admin, wp-include, etc)
/second_site/index.php
/second_site/wp-?????
/third_site/index.php
/third_site/wp-?????
None of the above is giving WP it’s own sub-directory.
/index.php
/wp-????? (a complete wordpress site, all your wp-???.php files, wp-content, wp-admin, wp-include, etc)
/second_site/index.php
/second_site/wordpress/wp-?????
/third_site/index.php
/third_site/wp-?????
In your web panel you’ll point a domain at /second_site and a different domain at /third_site or you should be able to get to those sites as:
domain.com/second_site/
Does that help?