IMAP works with folders on the server. That's the fundamental difference between IMAP and POP. Multiple client devices using IMAP will stay in sync with each other and with webmail, showing the same folders and messages (and the same read/unread state).
Apple's email clients will consolidate Inbox, Sent, Drafts, Junk, etc. It sometimes gets confused by naming variances and you'll have to manually mark the special folders.
Sent
Sent Items
Sent Mail
Sent Messages
Spam
Junk
Junk E-mail
Bulk
Deleted
Deleted Items
Deleted Messages
Trash
With the special-purpose folders recognized, Apple email clients show consistent aliases, regardless of the folder's actual name on the server.
I don't know if Thunderbird lets you display a different alias for a folder.
You might be able to rename those folders. The webmail system for each account might even have options to define the special purpose folders. Any changes you do there might cause error messages in an IMAP client until you refresh/reset/update the folder list (trying to refresh the contents of a folder name that doesn't exist anymore, for example).