Analogy: how do we know that the surface of the Earth is curved? Well, we could e.g. draw a triangle on the surface of the Earth, and check the sum of the corner angles. If the Earth was flat, you'd always find that the sum of these angles was 180°, so it would be impossible to e.g. create a triangle with two 90° corners. However, since the Earth is curved, this is indeed possible; you could e.g. draw a triangle where one edge follows the equator, and the two others follow meridians from the equator to the north pole.
The same concept would apply to spacetime: simple geometric relationships such as e.g. the sum of corner angles in a triangle would be different in flat spacetime from curved spacetime, and these relationships should be possible to measure to figure out the curvature of spacetime itself.
In a straightforward sense, spacetime is curved if the Riemann curvature tensor does not vanish. This is as mathematically objective as it gets.
Conceptually, how can we measure this? As mentioned in another answer, one way is to let a bunch of objects fall freely. Notice you don't use a single object, but a collection of them.
The reason is that locally (i.e., at a single point) you can't distinguish gravitational effects from acceleration. This is often exemplified by the thought experiment in which you can't distinguish whether you are standing on a rocket that is at rest on the ground, or if you are standing on a rocket that is accelerating in space.
However, gravity and acceleration are different when you consider effects at more than one location. For example, drop a bunch of apples around the Earth: they will get closer as they fall because they are all falling toward the center of the Earth. You get tidal effects, which are a hallmark of gravitational effect. The same thing would hold for light.
When we speak about curved spacetime does the definition take into account only the behavior of light in terms of its trajectory?
No. Spacetime curvature is a property of spacetime, not of the objects that move on it. It affects light, apples, oranges, and everything. While light does move in a slightly different way on spacetime (namely, on null geodesics, rather than on timelike geodesics), the properties of spacetime are the same for everyone.
In other words, if we take into account only the trajectory of light and not $m$, can we claim that the space-time on Earth is Euclidean and thus the space-time on earth is not curved?
The Earth also bends the path of light. While the effects are more subtle (because spacetime is less curved than near the Sun), they are still there.
Best Answer
Earth's gravity is quite weak and has essentially a negligible effect on the shape of the stick. Sticks (and basically everything else) are held together by electromagnetic forces and the Pauli exclusion principle, which are the same far from Earth.
Let's say you took the stick to vicinity to a small black hole, so the tidal forces (which are a measure of spacetime curvature) would overcome the strength of the electromagnetic forces holding the stick together. To counter this, maybe you apply some shear forces to the stick to keep in from bending. If you then remove the black hole but keep the shear force applied, the stick will bend. So you are right that gravity can deform an object, relative to its shape in the absence of a gravitational field.
A more "realistic" example of this is a neutron star in a binary. When two neutron stars orbit each other, the gravitational field of one can deform the other. This "tidal deformability" is imprinted in gravitational waveforms and can be used to learn about the internal structure of neutron stars such and the nuclear equation of state.