Sending messages with the Send as and Send on behalf rights
If User A has the Send as right to the User B's mailbox, and when sending a message User A enters the User B's email address into the From field, the message will be considered as sent by User B. Therefore, the rules covering User B will be executed.
However, if User A is defined as a Delegate of User B (the On behalf of permissions) and when sending a message User A enters the email address of User B into the From field, the message will be considered as coming from User A, and the rules covering User A will be executed.
Please note that the Sent Items Update (SIU) feature works only with messages sent using the Delegate (On behalf of) permissions. This is because the software always looks for messages to update in the Sent Items folder of a person who appears to be the sender.
When sending as, User A sends an email but it appears to be sent by User B. However, User B does not have this message in their mailbox. Therefore, the SIU service will not find it and will not update it.
In the case of send on behalf, a message is sent by User A on behalf of User B. The software still recognizes it properly as sent by User A and will update the mailbox of User A, as expected.