The thermodynamic definition of entropy of a system is
where T is the absolute temperature of the system, dividing an incremental reversible transfer of heat into that system (δQrev). (If heat is transferred out the sign would be reversed giving a decrease in entropy of the system.)
You may consider a vacuum as a system under study, but in this definition of entropy since there is zero over zero the change in entropy is undefined , so there is no meaning to "entropy of vacuum". Heat and temperature characterize large ensembles of particles, and also entropy.
you say :
I get confused a lot when we talk about a system. Most of the time, we seem to mean a fluid within a container (thermodynamics),
A system is what we have under consideration for an evaluation of thermodynamic quantities. Systems can be open, or closed/isolated; if energy can be exchanged with other systems they are open. The law of increasing entropy holds for closed or isolated systems.
In practice, when considering a problem, we can have approximately closed systems if one makes sure that the energy exchanges with the environment are minimal.
but what about the actual container itself? Can I make a claim about the entropy of an empty container before/after I fill it with something?
The container if it has a complete vacuum has no definable entropy. Once you introduce material you can have entropy.
I find conceptually more useful the definition of entropy in statistical mechanics, , where it is connected with the number of microstates in the system and can be shown to be identical to the thermodynamic entropy definition.
The macroscopic state of a system is defined by a distribution on the microstates. The entropy of this distribution is given by the Gibbs entropy formula, named after J. Willard Gibbs. For a classical system (i.e., a collection of classical particles) with a discrete set of microstates, if E_i is the energy of microstate i, and p_i is the probability that it occurs during the system's fluctuations, then the entropy of the system is
The quantity k_B is a physical constant known as Boltzmann's constant, which, like the entropy, has units of heat capacity. The logarithm is dimensionless.
In this definition it is again obvious that a complete vacuum has no definable entropy.