There is no God But Allah

We Muslims say “La Ilaha Ilalah” which translates to “There is no God but Allah”. However, in Arabic, Allah means God. Some also say, “There is no God but God”. Would it be better to say “There is no God But Allah” to signify the name Allah applying to the name of God? Because saying the latter doesn’t really make much sense. If the former, that doesn’t make that much sense either because it ultimately translates to “There is no God but God.” Does the word Allah mean God but does it also mean the name of God? Which is why it’s said: “There is no God but Allah.” Perhaps the Arabic phrase would get a better context?