Are they all prawns caught in the rivers of Ayutthaya?
You cannot tell from the dish name alone. If you want to know the actual origin and whether they are farmed, ask about the prawns you are going to order.
Grilled River Prawns · กุ้งแม่น้ำเผา
If you are looking for a meal to share in Ayutthaya, choose grilled river prawns. Prawns are grilled and the meat is picked out to eat, and it suits a group eating with rice and other dishes. The price differs with prawn size and selling unit, so look at the prawns you will order first and then choose.

It is a dish in which you enjoy the aroma of grilled prawns and the texture of the meat. Try the meat alone first, then add a little of the sauce served with it. Eaten with rice, you enjoy the flavour of the sauce too.
Ask whether the selling unit is per prawn or by weight.
River prawns are a well-known food ingredient on an Ayutthaya trip. If boat noodles are a one-bowl meal, grilled river prawns can be chosen as a dish to share with your companions. You need not assume from the name river prawns alone that they are wild or caught in the rivers of Ayutthaya.
You cannot tell from the dish name alone. If you want to know the actual origin and whether they are farmed, ask about the prawns you are going to order.
It depends on the selling method, such as prawn size and per prawn or by weight. If you get the price for the size and number you chose before you order, it is easy to settle the cost of the meal.
If you want to read more about these dishes and regions, these sources are a good place to start. Ingredients and cooking methods can differ from one eatery to another.
Sharing image credit: Takeaway · CC BY-SA 3.0