Two separate discussion going on now. One about yaw (or not) during a turn, and the other about sideslip (or not not) during asymmetric flight.
My view:
1. Yaw vs turn.
At 0deg AoB the a/c can be turned using 'all rudder' ie a flat turn. In this case the a/c is yawing but not pitching.
At 90deg AoB the a/c can be turned using 'all elevator'. In this case the a/c is pitching but not yawing.
The a/c can't instantly switch from one case to the other. It's a smooth change from one condition to the other, with yaw reducing & pitch increasing as AoB increases.
2. Slip vs asymmetry.
I think we all agree that in a wings level steady heading asymmetric condition there is some amount of slip. The Rel Airflow will be from the 'dead side' as a result of the rudder/fin sideload.
If an a/c is banked it will start to slip in the banked direction ie the relative airflow will have a lateral component (relative to the a/c axes) from the low wing side. This still applies in the asymmetric case. Imagine an asymmetric a/c at a very large AoB. It *will* start to slip towards the low wing. The slip will then result in a yaw towards the low wing. If the low wing is the live engine side then there will be a sufficient AoB that will counter the thrust asymmetry as a result of the relative airflow approaching from the live engine side causing a yawing moment towards the live engine.
It's possible to have sufficient bank that NO rudder at all is needed - ignoring some incredibly high thrust engine that could make the a/c do cartwheels...) albeit at a very high cost in drag. Not much good for performance then. On the hand, Vmc would be nice & low since the rudder hasn't reached its maximum deflection. In fact, it hasn't moved at all... Large angles of band are good for the manufacturer when it comes to certification since many things are related to Vmc so a low Vmc is 'useful'. Not so good for the pilot since it means unusual handling is required - hence the 5 deg AoB certification limitation.
Note the difference in the two cases above. Case 1 (wings level, only rudder used to oppose the asymmetry) the Rel. Airflow has a component from the 'dead side'. Case 2 (AoB, no rudder used to oppose the asymmetry) the Rel. Airflow has a component from the 'live side'.
As AoB increases towards the live side the component from the dead side must reduce until there is no sideslip, then increase again but this time from the live side.
So, it is possible to fly asymmetrically with zero sideslip and this AoB is where best performance will occur. This typically occurs at ~2-3 deg AoB. Any more than this and there will be a small sideslip towards the live side, any less than this and there will be a small sideslip towards the dead side.