what does it mean when experimental physicists refer to an experiments sensitivity? What in general is it compared to (e.g. observation?) Is there a formal definition or does it widely depend on the experiment in question?
Sensitivity refers to the magnitude of change of a measurement outcome for a given change in the measurement input. If a measurement is said to be “sensitive” to some variable, that means changes in the variable will result in changes to the output of the measurement; i.e. you can use the measurement to detect changes in the variable.
There are infinite examples, but consider for now the measurement of a volume of water. You want to pour water into a container, measure its depth, and from that figure out how much water has been added. Compare performing this measurement with (a) a pint glass vs (b) a bath tub. Clearly, the pint glass will give you a much larger sensitivity because the amount the depth changes when you add some more water is inversely proportional to the surface area. If you add a cup of liquid to the bath tub, you won’t be able to tell the difference, but you’ll see a huge difference if you add the cup to the pint glass instead.
But now you see that there are other considerations as well. If you measure the depth a bunch of times, how consistent are your results (“precision”; related to noise, maybe about the same for the bathtub and pint glass if you’re careful)? How much water can my container hold (“dynamic range”; bathtub can measure a larger change in water volume)? How well do I know the dimensions of the container to calculate volume (“calibration”; probably better for the pint glass)? Are there essential problems you need to overcome or correct for the measurement to be accurate (“systematic error”; maybe the tub has a leaky drain, which skews the results)? Etc. The sensitivity and the precision come together to give you a signal-to-noise ratio, which is a primary factor determining how reliably you can make conclusions.
Ultimately, experimental science is a practical thing, and there can be different definitions for these terms depending on the context.