Is this POP or IMAP?

It is POP

Since POP has no access to the server, I believe you will also need an IMAP account so you can view messages on the server.

Ok. Let it be IMAP. But how? How to delete all messages from thunderbird but also delete specified messages from server?

The challenge is that your desire goes against both POP and IMAP, so you need both. Set POP to leave on server, and then use IMAP to selectively delete.

As I said, I could do it with Mac's Mail and with Outlook Express.

You still didn't told me how to do it in Thunerdbird.

1. Let's say I have messages from example@example.com. 2. When message contains word "dinozaur", I want that message to be deleted from thunderbird and from server. 3. When message contains word "bike", I want that message to be deleted from Thunderbird but... left on server.

?

POP is designed to download and delete from server, while IMAP is designed to sync to the server. To make this work as desired, you need to have POP to leave on server, delete the dinozaur in IMAP account and the bike item from POP account.

I still don't get it :) 1. Do you mean I need to create two accounts. With the same name, but with different parameters (IMAP, POP)? 2. And then what? First I click download from one account POP?....IMAP...no... I don't get it :D

Could you write this step by step, please? What I click (in what order)?

1. create the IMAP account alongside the POP account
2. set the POP account to leave messages on server
3. use the POP account for routine email processing
4. for your example, let's assume you just downloaded the dinozaur and bike messages
5. for the dinozaur message, since you want it completely gone, delete from POP account and also from IMAP account
6. for the bike message, since you want it retained on server, just delete from POP account.

One more time :) By your solution. I have a list of messages. Each message is in two copies (from POP and from IMAP).

bike (IMAP) bike (POP) dinozaur (IMAP) dinozaur (POP)

I want ALL OF THIS MESSAGES to by deleted from thunderbird. So I have to on EACH MESSAGE click - delete, right?

If I do that, then from POP server: bike will be deleted and dinozaur will be deleted, right? And on the IMAP server: bike will be not deleted and dinozaur will by not deleted, right?

You are managing primarily from the POP account, in that all messages are left on PC and on server. To delete from PC only, just highlight all messages to remove and press delete key. So, highlight both the bike and dinozaur messages in POP account and click delete once. Since everything remains on the server by default, you just delete the dinozaur from server in IMAP account.