Kansas...was looking at some older forum issues. I'm guessing that your messages got rolled over to your phone answering machine because COX was recording a message at that time. Since you are primary on the Cox Voice, a busy signal where you were either talking or a message was being left, caused to roll over to your phones answering service. Mine is set up just the opposite, so I get an occasional Cox Voice mail and don't know till I check the Web Voice Tools.
As always CurtB, you are a wealth of info. I will pay attention when talking on the phone. Normally when on the phone, I hear a beep, beep indicating someone is calling. I either flash over to them if I notice a caller I know (when looking at the headsets display) or just let it ring and die. If at this time the caller chooses to leave a Cox voice mail, I'm not sure if the "stutter tone" will sound. Is the stutter tone different from my beep, beep, which is slow?
The beep beep you're referencing is the call waiting indicator (while you're currently on a phone call). If you have a new Cox voice mail, you'll hear a quick dut, dut, dut, dut... tone when you first pick up the phone receiver (prior to making a call) instead of the normal steady dial tone.