Documentation of Session extension incomplete

Preflight checklist

• I could not find a solution in the existing issues, docs, nor discussions.
• I agree to follow this project's Code of Conduct.
• I have read and am following this repository's Contribution Guidelines.
• I have joined the Ory Community Slack.
• I am signed up to the Ory Security Patch Newsletter.

Ory Network Project

No response

Describe the bug

If you extend a kratos session using the admin API,
the next call to the public toSession()/Whoami REST API will then return a new cookie.

This new cookie must be returned to the UI and used for future UI requests;
I believe the cookie contains an expiry data as part of its encrypted payload.

This is not explained in the kratos documentation and if this is not done,
UI requests start to fail when the original session expires even though the
session in the database has been updated.

Reproducing the bug

Not really relevant

Relevant log output

not relevant

Relevant configuration

not relevant

Version

kratos 1.1.0

On which operating system are you observing this issue?

Linux

In which environment are you deploying?

Docker Compose

Additional Context

No response

Start by locating the Kratos documentation for the admin session extension API and the public toSession()/Whoami REST API. Document that the refreshed cookie must be returned to the UI and used for subsequent requests, and confirm that the explanation covers session expiry behavior.