The documentation entry 1 at https://maven.apache.org/plugins/maven-site-plugin/faq.html is incorrect.
Calls the site phase of the site lifecycle, which consists in the following life cycle phases: pre-site, site, post-site and site-deploy. See Lifecycle Reference
Actually site will only execute the phases pre-site and site but neither post-site nor site-deploy. All the four phases are only executed if you execute site-deploy.