I would like to have a list for friends. The connection must be confirmed by the friend and should be entered on both sides.
If the connection is deleted from one side, the other should also be deleted.
That should already be the case by default. If it's not for you navigate to CB > Configuration > Connections and set both "Mutual Consent" and "Cross Connections" to "Yes".
Another list for colleagues and acquaintances, which does not have to be confirmed and should also be registered only one-sided.
CB does not provide multiple connection managers. You could maybe using the Following feature in CB Activity to output a follow button on users profiles so they can just follow other users instead of connecting with them.
Currently, the setting is the same for all types of connections. When creating the connection, I can only select a text for the type.
Correct, at this time there's no functionality for users to create their own connection types. They are admin defined until we rewrite our connections functionality.