Normally, if you set an outbound caller ID against an extension, (that extensions DID number), then that is what will be sent, as long as your provider allows this. If an extension calls using an outbound rule, with a caller ID set in that rule, then that caller ID will over-ride the one set in the extension settings.
There are some caller ID based options, in the outbound trunk settings, that, if changed can affect how this works as well.