With cold engine, reservoir full, while radiator less than half

The description sounds like a faulty radiator cap. The cap is actually a somewhat sophisticated part with two valves in it. It works as follows, You start the engine up with the radiator full of coolant and the reservoir at the low cold mark. As the engine heats up the coolant expands, on older cars this was just dumped out the tube and onto the ground. This left you with a radiator that would be full only when the engine was warm. Then they added the reservoir system with the intent of providing a safe place to capture that coolant and reduce gasket and seal failures as well. Those would happen because just like the cap allowed the coolant out through the pressure relief valve in the cap (that is the larger spring in the cap) it also would generate a vacuum as the coolant and engine started to cool off. So now the reservoir would catch all of the "overflow" but now as the engine cooled off the small center valve in the cap would be drawn open, and that pulls the coolant back through the hose into the radiator until the vacuum generated is equalized and the radiator is now full. This also allows for better temperature control because the coolant can maintain a level as high as needed based on the engines temperature. So say the engine runs at 180 degrees on a cold day in the winter, but the guy driving through death valley in august is cooking the engine at the 225 degree point. In both cases the coolant will remain full in the radiator. The other thing this system does is help with corrosion by keeping air from getting into the cooling system by maintaining the coolant at the full level, that prevents a lot of the scale and crud from forming in the system.
In your case the pressure valve in the cap is working and the coolant is escaping into the reservoir, but the return valve isn't and the coolant isn't being drawn back in. That points to either the cap being bad or there is a leak in the reservoir that is not allowing the return of the coolant. An easy way to test which it is would be to use a small vacuum pump to apply a vacuum to the return line that connects to the radiator below the cap. If a small vacuum lets you pull coolant through then it's more likely a bad cap. If not there is a leak or the line on the overflow of the reservoir itself is acting like a check valve. I have seen that happen if a bug built a nest in the tube.
As to the replacement cap, I try to use either OEM or Stant to be sure the new one isn't an issue. The Delco part number is 10316828 or RC94