Useful measurements. The issue is that your Arduino RX input is not pulled low enough (rather than not high enough).
I had a look at the Sparkfun logic converter that seems identical to what you used and it relies on the low voltage (3.3V) input pulling the voltage down which then passed through a mosfet transistor.
Therefore if the TX output of your loop controller can't pull the voltage low enough it has no chance to do so through the voltage converter. It would be ok if it was two inverting common emitter npn transistors circuits but it is not designed like that.
I also checked the circuit diagram of the loop controller and it has two 1K resistors on the RX and TX lines at the TTL input/output the Teensy.
So I recommend that you keep the voltage adapter in line as you have now, leave the Raduino/Arduino circuits as-is and remove (short circuit) the 1K resistor that is in series with the TX1 pin of the Teensy in the loop controller. This should then allow for a better drive of your voltage converter input and consequently of the input of the Arduino. You can leave the 1K resistor that goes to the Teensy's RX line.
At present I strongly suspect that the issue is that the 1K resistor in the loop controller circuit, is connected in effect in series with the 1K resistor between the Arduino RX line and the USB controller, and it does not allow the voltage to swing enough and that explains the difference in the low value (TTL logical one) between a powered up and an un-powered Arduino.
Hope that makes sense.
All the best,
73, John (VK2ETA)