Is matter inside spacetime or part of spacetime itself?

[Watari]

I'm trying to understand the concepts of physics, but there's one thing I don't fully get...
If there is some matter with a spatial and temporal coordinate, let's say coordinate X, is the matter "in" that time and place, or "is" the matter that time and place?
In other words, does space-time refer to the medium in which matter exists, or does it refer to the matter itself that exists there and then? Is matter part of space-time or is it something separate that is inside space-time?

 

[Orodruin]

In special relativity, space-time is the background in which matter exists. In general relativity, it is no longer simply a background, but is also affected by the presence of matter. This does not mean that it is the matter, just that it is affected by it just as electromagnetic fields are influenced by the presence of charges.