Google will modify the sender to suit the primary account used to send the mail. So my guess is your SMTP setting for that account use the wrong sign in name. ie 2 and 3 have the same username in the SMTP settings

Thank you for a prompt reply. It's taken me weeks to get back to it myself.

All three accounts had the same smtp server, namely smtp.gmail.com. There is a list which repeats this three times. One is marked as (default). My second and third accounts were using that one. So I carefully arranged for each account to use a different one in the list and that fixed the problem.

It still feels like a TB bug to me. The index into the list is now different in each case but the text used for the smtp server is identical. It's like TB looks up the server at one point and then uses the index into the list of servers for a reverse look up later. Looks like the reverse lookup code would need to be revised to get the source address from a different place.

There is a link in the account settings (above the sever settings in the account list) that names the SMTP to be used for that account.

The description is what is displayed and it is used for nothing only to make is simple for humans to comprehend which is which, so change the descriptions to something meaningful to you. Thunderbird defaults to server names. All good until you have multiples of the same account. You can call them anything, but I tend to include the local part of the email address so I can know what I am looking at.

Once you rename the list so they make simple sense to you it is a simple matter to check which is used by which account.

IT is also important to know when Thunderbird uses the default. And that is when the folder selected on the left is not clear as to which account to use.

A reply or forward should use the account that was used to receive the mail. This falls down is you have mail collected from one account (say GMX) by another (say GMail). In those cases you need to configure an identity in the gmail account for the GMX mail so Thunderbird knows what details to use in replies or forwards.

Hi Matt,

Thank you very much for the extra feedback. I really didn't expect anything further.

My Thunderbird is running in Windows 10 and I'm not sure I know what you mean. Perhaps you mean the account names in the left panel of the images. There is also further text in the "Manage Identities" box and I don't know what that is about. It is different for each account so perhaps that's it. I have attached two screen grabs. Would you please be so kind as to indicate what you mean on one of them (or something). Thanks, Billy Mes

scroll further down that list to outgoing server (SMTP) it is in there that outgoing server information is managed and where you can edit the descriptions so they do not all say Googlemail.