Why does iPhone Music app show me "Apple Music" logo instead of my iTunes songs?
I've learned more ... I may have an answer to my question.

According to this article here ... when I told Siri to "Play Accept Yourself by The Smiths", she may have played that from Apple Music, not from my own library.

That would explain why "Apple Music" got turned on.

But I don't know why this would happen because I did not renew my Apple Music subscription after the trial last summer.

After ensuring the "Show Apple Music" setting was OFF, I powered off my iPhone.  Restarted, and it looks like I'm back to "iPod" mode in Music.