After examining Fitting's definition of a model again, I think you're right. Thank you for this because this confusion would've definitely come back to stump me later on. I'm still looking for a more intuitive proof, though.

@EmilJeřábek This makes a lot of sense! Although I think this should be caused by the fact that Fitting uses one "global" language in his book and doesn't even really define the notion of a language. I think the semantics themselves are equivalent (at least I hope so). Thank you!

@AndrejBauer Yes, sorry! The relevant chapter is "Additional first order results", the subsection "Compactness". You will be able to find many alternative formulations with the same weird condition there. Also, in my research I've discovered that Fitting uses an unconventional notion of a first-order model, so the chapter on semantics might also be important.