I went to a seminar and a side question was if a theory had an omega model, however from the context I could not deduce the exact meaning. Does an omega model have a general meaning in mathematical logic?
Could an omega model be an $\omega$-model where its domain is of cardinality $\omega$ or could it be furthermore a unique model for such a cardinality, i.e. $\omega$-model is a countable model of the $\omega$-categorical theory?
There seem to be several uses of the terminology of $\omega$-model. Please, feel free to update the list below:
According to Theory of Recursive Functions and Effective Computability (1987) edition Rogers defines on the page 392 an $\omega$-model to be a model of second-order arithmetic with the domain $\omega$ (or $\mathbb{N}$) where $0, 1, +, \times$ are interpreted in a natural sense.
From Asaf's comment, another possible meaning is in set theory where an $\omega$-model means a model of set theory whose natural numbers are the true natural numbers. One property of these models is that first-order statements about the natural numbers are absolute between the universe and the model. So, for example, if $Con(\mathtt{ZFC})$ is true in the universe (and it is, since there is an $\omega$-model, so there is a model), then it is true in every $\omega$-model. This means that $\omega$-models "know" about models of $\mathtt{ZFC}$ as well. So this is quite a jump from just any model of $\mathtt{ZFC}$.