The mechanism of a subscriber call and the "outsider" call is different.
When you dial from your extension which is a unity subscriber extension, unity recognizes your extension and puts the voicemail email into the recipient mailbox with your name. The account used is UnityMsgStoreSvc account.
When the call originates from a number not recognized by Unity, then the message is placed into the recipient mailbox using Unity_ account. It's likely that you have a permission problem where that account is not able to place a message.
The best way to determine the problem is to look at the errors/warnings in the application logs when the call is placed. Please post complete messages for the time that you had this problem - there will be more than one.