There are two different things described by the word 'Universe'.
The first - more frequently used by mathematicians than natural scientists, as far as I know - means, roughly, 'all that can be', including things that, for instance, are not considered within 'existence' (which is a proper subset of the Universe) at some specific point in time, but can exist, given a particular set of variables.
The second, sometimes called the 'observable universe', is the collection of everything that can be said to exist for a particular set of coordinates in the various dimensions.
Things may 'randomly, spontaneously pop into existence out of nothing' in the second sort of universe, but not the first.
I am talking about the physical universe, not some abstract concept. Also, be careful when you say "observable universe", because the observable universe is a subset of the entire universe. It is the part of the universe, with Earth at its center, that we're capable of observing, due to the limitation that nothing can travel faster than the speed of light. See this if my explanation makes no sense.
The observable universe is a proper subset of the 'abstract concept' universe, as you put it. The observable universe cannot be considered a proper subset of the physical universe, but, rather, the
entirety of the physical universe (for the corresponding observer), as no information can be conveyed to the observer from beyond the boundary.
I don't see what I need to be careful about, as your definition agrees with mine: the collection of everything that can be said to exist for a particular set of coordinates in the various dimensions (e.g. with observer being on Earth at time x).
Things that exist further down the time axis ('future') are not considered a part of the observable universe for the corresponding observer (i.e. 'does not exist'), but are considered part of what you call the 'abstract concept' universe.
