What is the account type of the Gmail accounts on the laptop computer - POP or IMAP?

all three accounts are POP on both computers

With POP the default behavior is to download messages to your computer and remove them from the server. So when reading messages with your laptop computer, Thunderbird on the desktop computer would never see them. To change this you can set the option to leave messages on the server in your account settings for the POP account on the laptop.

I have it set that way already. All three accounts on both computers are set to "Leave messages on server" and "Until I delete them"

I was just looking at the gmail settings webpage for another question, and noticed there is a setting specifically for what gmail should do in a POP-connected account. There is an option to keep a copy on the server; is this set thus on yours?

Yes. In Thunderbird, all three accounts on both computers are set to "Leave messages on server" and "Until I delete them".

Go to the Gmail website in your browser. Open your Gmail account there. Open settings then pop/forwarding and look at the options offered for POP.

Thanks for your response. When I go to the gmail settings, under the heading " When messages are accessed with POP", "Keep Gmail's copy in the Inbox" is selected. Is that not the correct selection ?

I found the answer to my problem at

https://support.google.com/mail/answer/47948?query=recent&topic=&type=f&ctx=search

Thanks to "christ1" and "Zenos"for your assistance.

It is interesting that "Recent Mode" is in violation of the RFC's for POP mail. When asked by a pop client for all mail that is what is supposed to be supplied. Unfortunately for Google they have made such a mess of email that complying with the RFC would mean some clients would get 10 or more Gb of mail because now we save everything to the cloud and in Google's case they encourage permanent storage. Even it we don't want to. Good one Google, perhaps you to could call it "innovation"