Aha! The explanation makes sense. Thanks you.
I'd still like to see (or be pointed to) more general discussion of the fundamental difference in structure between G-mail and Apple Mail. G-Mail assumes one message may have many different filing attributes. Apple Mail assumes that a message will be stored in one folder (or would have to be duplicated if it were to be in multiple folders). This difference in structure seems to be at the root of the problem I was experiencing. But I would like to find advice on how to configure Apple Mail and/or G-mail so that Apple Mail can be used as the client for reading mail from a G-mail account.
(Both Apple and Google seem to love the spurious simplicity of providing minimal descriptions, and leaving it for the user --or the user community-- to grope their way by experiment, rumor, or superstition into finding best practice.)
Gmail is non standard IMAP. All email clients like Mail, Entourage, Outlook, Postbox, MailMate and others have to try and workaround Gmail's oddities. Each email client deals with Gmail in slightly different ways, but they are all trying to deal with Gmail's non standard IMAP.
If you use Gmail as POP you wouldn't have this issue, but you also loose all the advantages of IMAP.
Gmail wants you to use your browser to read your email. This way you get their ads and they can follow you online. Gmail is not going to change how it's setup so it works in your email client of choice. You can decide if you want to deal with Gmail's oddities or move to another more standard IMAP email. iCloud obviously works well with Mail as does Outlook.com and Yahoo. All free acounts come with some limitations. You get what you pay for.
I use MailTabPro for Gmail $1.99. In Mac App Store (MAS). It's simply a browser-like view of your Gmail account without having to open a browser window. I don't have to download anything to my computer. However if Gmail were to "loose or delete" my account, I have no backup of the messages on that account.
Here are some articles you might find enlightening.