On Thu, Sep 17, 2020 at 02:50 PM, Gene wrote:

I should add that it isn't clear to me whether this is a characteristic of GMail or of using two different protocols in general with providers.

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The latter.  Mixed access methods invariably lead to "unexpected results."  And even using POP3 and only POP3 on multiple devices often leads to a complete mess with regard to keeping things in sync if used with the default settings, which immediately removes a message from the server after download.
 
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Brian - Windows 10 Pro, 64-Bit, Version 2004, Build 19041  

Always remember that computers are just glorified light bulbs - they rarely fail in continuous use and usually go pop when turned off and on.

        ~ Technician with the username Computer Bloke, on Technibble.com