this is a naive question from a non-physicist. It is my understanding that gravitational waves are a deformation of spacetime. However, those are noticeable through, for example, laser interferometry. Then I cannot understand how these waves are anything but a variant of electromagnetic waves, i.e. completely generated and detected inside spacetime, without any alteration of the container itself. How can spacetime changes can be noticed from objects that are aware of nothing but their relative interactions with other objects? In other words, if I stretch a napkin on which I have drawn a grid, how does that make a change to the coordinates inside the grid? In this example, it obviously does not, as the coordinates depend on the grid, not something containing it. If I am not clear enough, I will reformulate.
Quick explanation for the part on gravitational waves would be the following. When one measures something, one always needs a reference, something to compare to, a ruler so to speak. Simplifying things a bit, to measure gravitational waves one uses two laser beams. Vaguely speaking, one compares one with the other. We expect gravitational waves (due to their polarizations) to stretch space in a certain direction but not both (except for subtleties with blind spots). In this way one laser beam will travel longer/shorter in the presence of a gravitational wave while the other remains unaffected, one compares them and concludes space has been perturbed. You can find more details by reading about interferometers.
About the statement saying that they are variants of EM waves, I can say well both phenomena are waves so they are both described by a wave equation, however their nature is very different. EM waves are the things we use currently to measure gravitational waves (i.e laser beams) by means of interferometers, but their polarizations, radiation (methods), mediator particle, spin, etc. are all different. So one has to be very careful when defining an EM wave, this sentence
"completely generated and detected inside spacetime, without any alteration of the container itself."
would imply almost everything is an EM wave.
Now, to address the other part about measuring geometry of space-time while inhabiting space-time, I will point you towards topics but will otherwise be very brief. What follows doesn't pertain directly to gravitational waves but to general relativity.
The point is understanding that certain geometrical properties are indeed recognizable at a local level. That is a surface (more generally a manifold for space-time), can be described locally by means of its metric which is the tool that lets you measure distances and angles at every point (this without the need to imagine your surface as being embedded into a bigger space). The metric then allows the curvature to be computed. The curvature tells us whether our surface locally looks like a sphere or like a saddle or if it looks flat and so on. So I guess your question somehow reduces to how do we measure the curvature of our space-time manifold as inhabitants of said manifold. There is no one way of doing this, but just to mention some ways I will say, by parallel transporting vectors, that is tracking how a certain fixed direction changes while you follow a closed path in your manifold. Another option involves bundles of trajectories called geodesics, where the curvature is related to how they spread from one another while travelling. These are just geometrical facts upon which general relativity is built on.
Your example with the napkin is very specific and enjoys a lot of symmetry, so it doesn't really serve to counter your intuition on measurements. If you expand a napkin with a grid (with which you are probably thinking about the universe expanding) you are implicitly saying that your objects are fixed to the grid which is not how we would know the napkin is expanding. It is true that spatial features like distances and angles will remain the same (see conformal transformations), that is things that are fixed to the grid (comoving objects) won't notice any change in relative distances or angles. But about things that are not fixed to the grid (like light beams) we can say a lot, think about using the Doppler shift for measuring receding/approaching speeds.
If you are really interested you might want to learn and read more on differential geometry and general relativity at the formal level. For which you can start in Wikipedia and then move on to the references therein. I hope this brief explanation serves its purpose.