Applying compactness theorem to show a property of a theory $K$

Question

First of all, the statement that I'd like to prove is basically similar to the statement in these three links below:

1. Prove there is no theory whose models are exactly the interpretations with finite domains
2. Compactness and axiomatisability
3. compactness model theory question

Let me write down what the statement that I want to prove: There is no theory $K$ whose models are exactly the interpretations with finite domains. In all of these three links, they use compactness theorem and they consider a wf $$\varphi_n=\exists x_1\cdots\exists x_n\left(\bigwedge_{0<j<i\le n}x_i\ne x_j\right).$$

I kind of understand their proofs, but I do have a doubt about these proofs. Can I really consider the above wf in a general theory $K$? It seems like I can consider that wf when $K$ is a first-order theory with equality.

If I can't consider the above wf in a general theory $K$, how can I prove the statement? I'd like to have an idea how to show it, so a hint would be appreciated.