Does MySQL support defining a table inside a session without appending it to the database?
You are right it is a like a local table to the connection.
According to the link you have under the subheading Temporary Tables, it says:

A TEMPORARY table is visible only within the current session, and is dropped automatically when the session is closed

Simply put, once the DB Connection terminates (normally or abnormally), the table disappears.
Also, please be aware that only the DB Connection that created the temporary table can access it. If 10 different DB Connections perform CREATE TEMPORARY TABLE, each DB Connection would have its own table. No other DB Connection can access it.