Wall Heater in 1930's building

Hello,

I live in an apartment built in 1928. The heat is provided by in wall electric heaters. The main heater is basically on or off. What I am trying to do is wire in a thermostat. When I take the heater off the wall there are three wires coming from inside the wall... Black, white, and uninsulated. The heater itself has two wires, black and red. It is currently wired black from the wall to black of the heater, and white from the wall to red of the heater, and uninsulated is screwed to the chassis of the heater. The thermostat I'm trying to use is just an on off switch which can be controlled over the internet.

Ideally I would like to take the wiring from the wall into a three prong electrical receptacle, plug in this internet switch, then go out of the switch onto the black and red wires of the heater.

When I measure voltage from the wall between the black and white wire I get 240V, when I measure between the uninsulated to the black or white, I get 120V. When I wire the wall wiring to the three prong outlet, I place the uninsulated on ground, the black to the hot, and the white to the neutral. To test it, I'm using one of the basic three prong electrical testers with the red and yellow lights on it. Performing this test it shows hot and neutral reversed. When I flip flop black and white, I still get hot and neutral reversed. The only way the tester shows the wiring as being correct is when I wire the outlet ground and the outlet neutral to the white wire from the wall and the outlet hot to the black wire from the wall. With it wired that way, and I measure voltage across the hot and neutral off the outlet I get 240V, which will probably fry the internet switch.

How do I wire this thing so that it will be safe, or is it even possible? If it is not possible, was the heater wired correctly to begin with? Heater black to wall black, and heater red to wall white, and uninsulated to heater chassis?

1000 thank you's in advance for any help!