If space is empty of matter, how can a vacuum - devoid of matter - exist in that emptiness? It's as though they've given vacuum a three dimensional form, just to define it as nothing.

Outer space actually isn't a vacuum - on average there is about one atom per cubic centimetre of space. That certainly still is a lot of empty space though. It may be even more surprising to learn that even our atmosphere at sea level is 99% empty space. So the question of what exactly 'space' is is a great one.

We gain more insight by looking at Einstein's general theory of relativity. This posits that we are enmeshed in four-dimensional 'space-time' - the three space dimensions we know and love, with time being the fourth. Space-time is curved by the presence of mass, such as planets and stars, or even single atoms or subatomic particles. This curvature then has an effect on all kinds of matter and energy within it. General relativity even suggests that the overall shape of space-time in the universe may be one of theee shapes - flat, closed like a sphere, or open like a saddle. Observational data so far suggests a flat universe, but other shapes haven't been ruled out either.

In this view, space-time is just a mathematical framework that accurately describes what we have observed in the universe so far, and whether or not it is a real tangible thing is more of a philosophical question. Perhaps it is just an emergent property from the interaction of other entities, or just a handy model, without really existing. Similar debates exist about the nature of mathematics. Ultimately it doesn't have any practical implication - the model of space-time allows us to explain what we see and whether or not it actually exists is irrelevant. For practical purposes, we can behave as if it does exist.

As an extra aside, even in a complete vacuum, 'virtual particles' - pairs of a particle and its anti-particle - are predicted to appear and disappear frequently as a consequence of the mathematical models of quantum mechanics, meaning there is no such thing as a true vacuum.

I hope this wasn't too complex or detailed - there are many wonderful books dedicated to explaining these concepts simply but in much more detail than I could here - A Brief History of Time by Stephen Hawking is a great start.