You're using the pipeline (|) to provide input to Out-File.
Only stream 1, the success (data) output stream is sent through the pipeline by default.
The conceptual about_Redirection help topic, from which you quote, documents all 6 PowerShell streams.
Optionally - as you have done - you can merge other streams into stream 1, which causes their output to be included as well.
You can merge:
individual streams by their number; e.g. 3>&1 merges the warning stream (3) into the success stream; multiple such redirections may be used.
all (other) streams, represented by *, i.e. by using *>&1
Since you are using *>&1, output from all streams is sent through the pipeline and therefore saved to the target file by Out-File.
As an aside: >, the redirection operator (along with >>), is an effective alias of Out-File, so you can use it alone to save to a file - assuming Out-File's default character encoding is acceptable:[1]
& { 'success'; Write-Warning 'warning' } *> allstreams.txt
Targeting files with > also allows you to save individual streams to separate files:
& { 'success'; Write-Warning 'warning' } > data.txt 3> warnings.txt
[1] In PowerShell version 5.1 and above you can change Out-File's default encoding in a manner that also takes effect for >, via the preference variable $PSDefaultParameterValues - see this answer.