What is meant by "Nothing" in Physics/Quantum Physics?

In Physics "nothing" is generally taken to be the lowest energy state of a theory. We wouldn't normally use the word "nothing" but instead describe the lowest energy state as the "vacuum". I can't think of an intuitive way to describe the QM vacuum because all the obvious analogies have "something" instead of nothing "nothing", so I'll do my best but you may still find the idea hard to grasp. That's not just you - everybody finds it hard to grasp.

Start with the classical description of an electric field (Maxwell's equations). It's not too hard to image an electric field as a field filling space. You can even feel the field: for example if you put your hand near an old style TV screen you can feel the static electricity. You can imagine turning down the electric field until it disappears completely, in which case you are left with the vacuum i.e. nothing.

Now imagine the same field, but this time we're using the quantum description of the field (Quantum Electrodynamics instead of Maxell's equations). At the classical level the field is approximately the same as the description Maxwell's equations give, but now we have fluctuations in the field due to the energy-time uncertainty principle. Just as before, imagine turning down the electric field until it disappears. Unlike the classical description, the (average) electric field may disappear but the fluctuations do not. This means the quantum vacuum is different from the classical vacuum because it contains the fluctuations even after you've turned the field down to zero.

The key point is that when I say "turn the field down" I mean reduce the energy to the lowest it will go i.e. you can't make the energy of the electric field any lower. By definition this is what we call the "vacuum" even though it isn't empty (i.e. it contains the fluctuations). It isn't possible to make the vacuum any emptier because the fluctuations are always present and you can't remove them.

At an even more abstract level (and inspired by the John Rennie's TV analogy): you seem to think of "Nothing" as the equivalent of a black TV screen. In modern physics, "Nothing" is similar to the noise between TV channels.

If we take "nothing" to be the same as "zero", "something" to be the same as "not-zero", the vacuum state is both "nothing" and "something".

The "nothing" part of the vacuum state as a theoretical object is that the average value of a series of measurements of the field will be zero. The "something" part of the vacuum state is that the value of any single measurement will in general not be zero. When we can't predict single measurement results and how they will vary over time, we often find that we can predict average values and how the average values will vary over time.

There will most likely be technical aspects to any Physicist's answer here. In the above, "measurements of the field" must be understood to have quite theoretical connotations. John Rennie has labored heroically, but ultimately you have to work at being an intimate friend of the Math and its relationship to experiment.

You seem to be trying to make "nothing" be something vaguely different from any mathematical idea.