how much ADC noise adds to signal?
But adding a 50 Ohm terminator to Ch 1 did not change n(t) at all, which makes me question my understanding.


Oscilloscopes are usually rather noisy, so n(t) may be dominated from internal noise in the scope itself - which is not affected by an external resistor at all.


  Question 1: What is M?


That looks a lot like signal-to-noise ratio.


  Question 2: How would n(t), the signal I observe on ch 1, change


Changing the termination can have "interesting" effects on a high frequency transmission line for example. 

Bottom line: You cannot tell without a schematic and also layout in high frequency cases.