"The database server must be shut down in order to get a usable backup."
Note: must is in italics.
Then we have section 25.3.3.1. Making A Non-Exclusive Low-Level Backup
which contradicts this statement with "Perform the backup, using any convenient file-system-backup tool such as tar or cpio (not pg_dump or pg_dumpall). It is neither necessary nor desirable to stop normal operation of the database while you do this."
While I understand this is a complex topic. I think at the very least we can refer to section 25.3.3.1 in section 25.2. I also think that the discussion about frozen snapshots is ill advised. If I were doing it I would use pg_start_backup before using any file system snapshot.
Another thing we've talked about on this topic before (but never got around to doing all the way) is to re-order things so that the current 25.3 chapter comes before 25.2. And making sure we put the simple yet correct solutions like pg_basebackup first, and put complicated ones later.
And we should really chance the title of what's now 25.2 to explicitly be called "offline file system backups" as well.