In Brit/Aust/NZ English and probably elsewhere, "takeaway" food is what is called "take out" in AmEng. So that's where the confusion comes in.
At a restaurant I would say: "We've finished, thank you" and the waiter should clear the table. If he hovers over some item he's not sure about you can help him with "Yes, you can take that".
Taking your remaining food home from a restaurant is often referred to as having a "doggy bag", the idea being that the food is for your dog, which occasionally it may well be. This term is very widespread.
If you want to say that you'd like to take it with you: "We've finished, could we have the rest to take home?". The waiter will say something like "Certainly, I'll get it ready for when you leave".. or he might even say "Sorry sir, we don't do doggy bags".
not a teacher
Student or Learner