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The Cavendish experiment was equivalent to measuring $G,$ Newton's gravitational constant. However, because physicists at the time did not write equations in the same way we do now, Cavendish didn't have the concept of $G$ and reported the results of his experiment as a measurement of the mass of the Earth.
When did physicists begin using $G$?
The constant was introduced in Poynting's essay The Mean Density of the Earth that won the Adams prize at the University of Cambridge in 1893 on the subject "The Methods of determining the absolute and relative value of gravitation and the mean density of the earth". It is introduced matter of factly on p. 2 of the essay:
"We are therefore led to conclude that the law is general, or that if we have any two bodies, of masses $M_1$ and $M_2$, $d$ apart, the force on either is $$G\frac{M_1M_2}{d^2}$$ where $G$ is a constant, the constant of gravitation.
He proceeds to describe experiments that determine $G$ (including density of the Earth measurements a la Cavendish). Later (p. 5) he also calls it "the constant of attraction".
The next year Boys made a bigger deal out of $G$ at a June 1894 meeting of the Royal Institution of Great Britain, with reference to Poynting's essay. He names it "the Newtonian Constant of Gravitation", reinterprets Cavendish's experiment as measuring it, and illustrates its significance with passion. Here is how it is introduced:
"Newton showed that to complete his law and to put in a numerical constant (the Newtonian Constant of Gravitation) that would convert his proportion into an equality, two methods are available: we may either make observations on the disturbance of the earth's gravitation by the action of isolated parts of it, we may either find the relative attraction of an isolated mountain or the strata above the bottom of a deep mine, or we may make an artificial planet of our own and find the attraction which it exerts.
The Newtonian Constant will be known if we know the force of attraction between two bodies which we can completely measure and weigh. Employing tho C. G. S. system of measurement, the Newtonian Constant is equal to the force of attraction in dynes between two balls weighing a gramme each, with their centres one centimeter apart. Of course it may be referred to pounds and inches or ton.a and yards, but as soon as all the quantities but $\textrm{G}$ in Newton's equation $$\textrm{Force} =\textrm{G}\frac{\textrm{Mass}\times \textrm{Mass}}{\textrm{Distance}^2}$$ are known, no matter in what units the quantities are measured, $\textrm{G}$ is known. The conversion of its numerical value from one system of measarement to another is of course a mere matter of arithmetic."
$G$ was invented after a system of units that included force was devised.
Jungnickel & McCormmach's Cavendish: The Experimental Life pt. 2, §16:
The Cavendish experiment today is often called the experiment to determine $G$, which is correct given that the experiment is the common possession of physics. It is often said that Cavendish’s object was to determine $G$, which as a historical statement is incorrect but understandable given that the constant is more significant than the density of the Earth. In Cavendish’s time, there was no independent unit of force, such as our dyne and Newton. The strength of any force was expressed in terms of an equivalent gravitational attraction, and weight was the measure of mass. The universal gravitational constant did not come up, though we can easily calculate it from Cavendish’s data.89 We find implicit in his work two of the three principal universal constants, the velocity of light $c$ and $G$ (Planck’s constant $h$ is the third), but Cavendish did not think of $c$ as necessarily having a constant value, and it was the better part of a century after Cavendish’s experiment before $G$ entered physics.
89. Cavendish did not write an equation for the force of universal gravitation, as we do: $F=\frac{GMm}{R^2}$. He could have calculated $G$ without having a unit of force, but he had no need for it, and it would not have occurred to him. Clotfelter (1987, 213).
Clotfelter, B. E. “The Cavendish Experiment as Cavendish Knew It.” American Journal of Physics 55, no. 3 (March 1, 1987): 210–13.
p. 213:
If Cavendish, or one of his contemporaries, had wished to calculate a gravitational constant, how could it have been expressed? The system of units in use did not include a unit for force; no unit of force was proposed until 1873, when the dyne was introduced. Cavendish expressed distances in feet and inches and weights or masses in grains. One might express a gravitational constant without using a unit for force, but surely the idea of measuring such a constant is less likely to occur to an experimenter when no unit for force is available.
We may conclude that Cavendish did precisely what his paper says—he measured the density of the Earth. His report of the experiment gives no hint that he thought in terms of gravitational constant, and his analysis neither needs nor suggests a constant in the equation expressing universal gravitation. It is, of course, not difficult to take the data Cavendish gave and derive from them a value of $G$, but he did not do that and he did not suggest that he knew that it might be desirable.
This is a very interesting question that I spent much time studying. Newton never used nor invented a gravity constant. The formula for gravity force he states by words corresponds to $F=Mm/r^2$, so without any gravity constant. It is then of great importance that Newton had a very different view on matter (and mass) than modern physics. He explicitly states that the philosophy of all his work is rooted in that matter consists of indivisible particles with spatial dimensions. Newton and other scientists used Newton's work for hundreds of years to do long series of predictions, such as relative masses between heavenly objects. Newton even calculated very accurately the density of the Earth relative to that of the Sun. What Newton struggled to do was to find the density of the Earth relative to a known uniform substance. The first one to do this accurately was Cavendish in 1798. But also Cavendish did not use nor invented nor measured any gravitational constant. What is true his measures as well as a Cavendish apparatus can be used to find the gravity constant, but that is first after one re-define mass in terms of kilogram.
Newton mentioned mass and weight are proportional to each other as long as the objects of interest are at the same distance from the center of the gravitational object. This is true also.
The gravity constant was first mentioned in 1873 by two French physicists. This I think is not coincident. The gravity constant is needed when one re-formulates mass as kilogram. The kilogram was invented in France just before the beginning of 1800, but was first accepted in scientific circles around Europe around 1870. In 1873 they used notation f for the gravitational constant, some soon used k. People in 1894 used the notation G, but it took many years before G became the standard notation. If one uses f, G, or k as a symbol, it is naturally just cosmetic. What is important that no one used the gravity constant before 1873.
I have tried to summarise some of the history of the gravity constant in a recent article (not in a top journal), but still perhaps of interest as I many references to work there that people themselves can look up and make up their own minds history and use of gravity constant.
