  The omitting types theorem goes back to the mid 1950s. It very definitely depends on the language being first-order and countable. It has several useful generalisations, for example model-theoretic forcing, which is an analogue of the forcing construction in  set theory.  In fact the games used for model-theoretic forcing (see the entry on  logic and games)  can be adapted to prove the omitting types theorem too. There are similar but more complicated theorems for uncountable first-order languages; some of these can be paraphrased as omitting types theorems for  infinitary languages.