In the context of Boomla, agent means the user agent, typically a browser, identified by an agent cookie. If the agent cookie is not transmitted in subsequent calls, the requests are considered to be coming from different agents.
If a branch is private, only its owner and users registered to the website can access it through a browser.
Note that registered users still don't have filesystem level access, unless explicitly granted by the owner.
Automated changes like system updates are performed by the System account, also referred to as the System user depending on context.