Are they all prawns caught in Ayutthaya's rivers?
You cannot tell from the name of the dish alone. If you want to know the real origin and whether they are farmed, ask while looking at the prawns you will order.
Grilled River Prawns · กุ้งแม่น้ำเผา
If you are looking for a meal to share in Ayutthaya, choose grilled river prawns. The prawns are grilled and you peel the meat out to eat it, which suits a group having rice and other dishes. The price changes with the size of the prawn and the selling unit, so look at the prawns you will order before you choose.

It is a dish where you enjoy the aroma of grilled prawns and the texture of the prawn meat. Try just the meat first, then add a little of the sauce that comes with it. If you eat it with rice, you can enjoy the taste of the sauce too.
Ask whether the selling unit is per prawn or by weight.
River prawns are a well-known ingredient on an Ayutthaya trip. If boat noodles are a one-bowl meal, grilled river prawns are a menu item to share with your companions. You do not need to assume from the name river prawn alone that they are wild-caught or caught in an Ayutthaya river.
You cannot tell from the name of the dish alone. If you want to know the real origin and whether they are farmed, ask while looking at the prawns you will order.
It depends on the size of the prawns and on the way they are sold, such as per prawn or by weight. If you get the price for the size and number you choose before you order, it is easier to plan the cost of the meal.
If you want to read more about the food and the regions behind it, these sources are a good next step. Ingredients and cooking methods can differ from one place to the next.
Sharing image credit: Takeaway · CC BY-SA 3.0