Are these messages being moved by a filter that is defined on the mail provider's site, before they are downloaded and filtered by TB? Some providers sort mail into 'Important', 'Promotions' etc.

Not at all. Like I said, for instance, if I place an order at a new website, instead of going straight into the main inbox as an unrecognized email, it goes to the one labeled "Charity Requests/Crusades," where all my petitions go. It's like this folder has taken over as the main inbox.

Who is the mail provider, and how is your filter set up? Post a picture of the filter.

Sure. The provider is gmail and here's a picture of one the filters (they're all basically the same - no filter/unknown email, it should go straight to the inbox and it doesn't, coming to the charity requests folder the second filter sends emails to).

HOWEVER, I have Thunderbird on another computer (also a Mac) and the inbox behaves normally with the same filters. This is a problem specific to Thunderbird on the laptop.

Thank you.

You could try using the FolderFlags add-on to see if this subfolder is mistakenly working like Inbox, or delete the folder, create a new one, and see if that makes a difference. Enable the filter log in Tools/Message Filters in case it provides a clue.

Gmail does have the option to sort mail into Primary, Promotions, Social etc., but that's not happening here, as long as there are no filters defined on Gmail.