Good question about the connector. One thing about BPMN is that within each pool, the process is as it appears to the owner of the pool. From the contractor’s perspective, he submits an invoice and gets paid. The same way that he would throw a ball up and catch it. The fact that there is something else affecting the sequence of events is irrelevant to the contractor. His process (as far as he knows) is to get paid, if he submits the invoice, without having to do anything else. The messages (dashed lines) between the swim lanes indicate that the process involves communication, and lets anyone reviewing the diagram appreciate exactly what it is that “someone else” has to do.

BPMN is good that way, in that you can view the process from either participant’s perspective, and overall.