The following sentences are from "An Introduction to Mechanics" Kleppner and Kolenkow. I couldn't really understand what the example (in italics) is trying to explain especially the sentences in bold :
Operational definitions may be satisfying theoretically, but they can
be useless in practice because in principle they are limited to situations
in which the operations can be carried out. However, this is not usually
a problem; physics proceeds by constructing a chain of theory and experiment
that allows us to employ convenient methods of measurement
but that are ultimately based on the operational definitions.For example,
the gravitational force on an object turns out to be proportional to its
mass. Because weight is proportional to mass, one can compare masses
simply by comparing weights.The most practical way to find the mass
of a mountain, for instance, is to observe its gravitational pull on a test
body, such as a hanging plumb bob, essentially comparing the mass of
the mountain to the mass of the Earth. If we had to employ the operational
definition of mass, we would need to apply a standard force and
measure the mountain’s acceleration. This would be impractical, to say
the least. Fortunately, the two methods are directly related conceptually.
Best Answer
Basically what they are saying is that there is a difference between how things are theoretically defined and how you'd measure them in practice. In this concrete example, the operational definition of mass seems to be the one in Newton's law: $F=ma$, i.e. the inertial mass, or resistance to acceleration. Based on this definition you would measure the mass by applying a known force to an object and measuring its acceleration.
Mass is rarely measured this way (e.g. in a mass spectrometer). More commonly we just put objects on a scale and measure the gravitational force (=weight) which is proportional to the mass. This is an indirect measurement.
As for the mountain, the only practical way to measure its mass would be through its gravitational force on another object of known mass.
So no really deep meaning here. Only that in practice we have to do things differently than in theory.
To give you another example,the metre is defined as the distance that light travels in 1/299792458 seconds. However usually you don't measure distances by stopping the time that light takes to travel.