Well it's a differentiation the locals seem to prefer too so I'm just rolling with what I was briefed, and also making a point about Arabic language so referring to god as allah in this case is 100% correct. I'm well aware they worship the god of Abraham, but differentiation between the three Judeic belief systems exist so if using the native term for god allows you to specify which edition you're talking about when discussing the concept of a higher power, then it makes perfect sense to use. The goal of language is to effectively communicate whatever the hell you're talking about in as few words as possible. From my understanding in the native tongue, MY gods would not be referred to by the term Allah or any of it's related terms, suggesting the word itself is restricted to I Am Who Is linguistically as well.
I took some Arabic linguistics so I'm familiar with the "casual" use of Insh'Allah in some Arabic speaking nations, the translations we both provided work out either way in English as at least here in the U.S. you see christians in particular do the same pattern of "leaving it over" or "surrendering" to god on whatever you're talking about. So yes slightly more accurate, but a hill of beans in the end.