Are they all prawns caught in the rivers of Ayutthaya?
You cannot tell from the name of the dish alone. If you want to know the real 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, pick grilled river prawns. The prawns are grilled and you peel the meat to eat it, so it suits a group with rice and other dishes. The price changes with the size of the prawns and the selling unit, so look at the prawns you will order first and then choose.

You enjoy the aroma of grilled prawns and the texture of the meat. Try the meat on its own first, then add a little of the sauce that comes with it. Eat it with rice and you taste the sauce too.
Ask whether the unit is per prawn or by weight.
River prawns are a well-known ingredient on a trip to Ayutthaya. If boat noodles are a one-bowl meal, grilled river prawns are a dish to pick when you share with your companions. You do not have to assume from the name river prawns that they are wild or that they were caught in the rivers of Ayutthaya.
You cannot tell from the name of the dish alone. If you want to know the real origin and whether they are farmed, ask about the prawns you are going to order.
It depends on the size of the prawns and how they are sold, such as per prawn or by weight. If you get the price for the size and number you chose before you order, it is easier to set the cost of the meal.
If you want to read more about the food and the regions behind it, these sources are a good place to start. Ingredients and recipes can vary from one restaurant to the next.
Sharing image credit: Takeaway · CC BY-SA 3.0