Is the fact that something is coupled to something else mathematically well-defined?
Sort of, because when you say that X and Y are coupled to each other, it means that X and Y interact with each other. This interaction is encoded in a mathematical way, and can appear in many places in different ways (in the action/Lagrangian, in the equations of motion, in the physical scattering amplitude, in the gravitational potential, etc.). I say 'sort of' because the exact mathematical expression which encapsulates the occurrence of the coupling depends on the quantity you're interested in studying. The fact that X and Y interact with each other is a physical statement, which has physical implications - a universe without interactions among any entities is boring because nothing is created, destroyed or modified. Fortunately, our universe is not like that, and hence we exist. X and Y here can be anything, but usually, they correspond to 2 different fields. 800
One counterexample is when people say that the <PERSON> scalar $R$ (not a field) is coupled to a scalar field $\phi$ in a scalar-tensor theory of gravity.
Do people use the term differently or does it always mean the same?
They typically mean that two entities X and Y, whatever they might be, interact with each other. What X and Y are, and what the mathematical expression for their interaction is, is usually clear from the context.
Usually, in a QFT, you need to perform perturbation theory to compute physical observables (scattering amplitudes). Amplitudes depend on the coupling $g$. Perturbation theory just means that you approximate the exact answer by writing only the first few terms of it by performing a series expansion in integer powers of $g$. Sometimes you need non-pertubative effects, and even there you will get a functional relationship of the quantity you're interested in in terms of g, e.g. the <PERSON> electron-positron pair production rate depends (schematically) on $e^{-1/g}$ where g = electric charge (coupling).
In gravity (GR), the coupling is $\sqrt{G}$, which is quite small (in SI units, for instance), which is another way of saying that one can approximate GR as a weakly-coupled QFT of gravitons (as long as the energy scales probed are smaller than the <PERSON> scale). This small coupling then gives the physical result that 2 electrons will repel each other due to electrostatic forces much more than they will attract each other due to gravity.
In short, a coupling is just a parameter, which tells you how much/how quickly/how slowly something is created/destroyed/modified/exchanges energy and momentum with its surroundings. 800