Connecting logic gate input pin with pull up resistor to another driver input pin doesn't keep the voltage
Look in your ULN2803A datasheet. You'll see that it contains a 2K7 base resistor into the Darlington transistor pair and parallel pull-down resistors to disable the transistors when the input is floating.

So an input voltage above something like 1.4 V (two base diode drops in series) is being loaded by the 2K7 resistor in series with two forward-biased diodes in series - 1.4 V-ish. Your 4K5 pull-up resistor provides the top half of a potential divider with the load 2K7 and diodes at the bottom, leading to your 4 V found.

Try using a driver with a high input impedance instead of the input current thirsty ULN2803A. Look for driver ICs, there are plenty far more modern than this part. Or you may be able to use an op-amp connected as a voltage comparator that transitions at 6 V. It's what suits your load current for it.