Should read

Unlike the other IMAP account (see image 2) I have the new one has nested subfolder (see image 1 & 3).

This might be fixed by opening Tools/Account Settings, Server Settings for the account, Advanced..., and changing the 'IMAP Server directory'. For gmail accounts, you would set this to [Gmail].

I see in the other IMAP accounts the 'IMAP Server directory' empty

However, I see that "Personal namespace" is set to "" in the non-problemtatic one whereas the in the problematic account it is set to "INBOX."

Should I set the "Personal namespace" to ""?

You can try it, but review the 4th paragraph here:

http://kb.mozillazine.org/IMAP:_advanced_account_configuration

Thanks - I will give it a read. Not sure I understand some of it