The way I make the call would be: "Del Rio Traffic............Del Rio"
I also tend to say what the aircraft is (i.e. Skyhawk [Tail-number exlcuding N]) Just to let the traffic know what type of aircraft it is and what to be looking for.
I do not see a problem with ending with Del Rio Traffic, if you wanted to get smart with the guy you could have said your pet peeve was people transmitting non-essential information over the traffic advisory freq.