As that documentation page states, all the software listed in "Installing additional software for Alfresco Community Edition" will / can be installed using the installer, and it is optional to install them manually instead of having the installer do it (they can be disabled during installing).
There are no concise / singular installation instructions for doing a proper, non-monolithic installation without using the installer in the documentation. There are various documentation sections that address parts of the install procedure, and it is up to the user / administrator to figure out what they need for their environment.
Angel Borroy did a talk at last BeeCon about installing Alfresco components one-by-one which you might want to check out.
Basically, for a non-installer setup you'd need to install/setup:
- Java JDK 8 (JRE is technically sufficient, but for some admin / support interactions, JDK is preferrable)
- Apache Tomcat 7 (ideally, you'd have 2 or 3 separate instances to separate Repository from Share and SOLR - the latter only if you don't use Alfresco Search Services)
- PostgreSQL / MariaDB / MySQL
- OpenOffice / LibreOffice (incl any additional fonts you may need to support your office file contents)