You SHOULD have a resistor. The value depends on the conductivity of the water. Pure water will exhibit a very high resistance across the probes but dissolved salts will make it much lower. You have to pick a value that ensures a large swing in voltage as the water bridges the metal tips. For example, if you use a 10K resistor and the water exhibited a resistance of 10K you would get half the 5V at the ADC input. You would get the same voltage if you used a 100K resistor and the water also measured 100K, it's the ratio that matters. Intuitively, I would suggest 10K is a good general value to choose.
Please remember this does not measure the amount of water, it only measures whether both probes are immersed at the same time. As your graph showed, you get two readings, one if both probes are immersed and another if one or both are dry.
You might get a reading with no resistor but treat it as suspicious, it would come from electrolytic action between the metals of the probe and dissolved elements in the water. The reading might be completely different from one water sample to the next.
Brian.