At what level do children become classified as chess prodigies?

There were prodigies long before there were titles. 

They stand out, becoming widely reputed at a ridiculously young age. 15 is too old. Under 10 seems more reasonable.

I would consider a prodigy these days be of titled player strength of 16 or less

Probably 14

Reaching 2000 elo before or at the age of 8 is pretty prodigal. It's all relative like the queen of Katwe, she's considered a prodigy in Africa but turns out around 1700 in the professional world. So I agree there needs to be a formal definition - even if that's a debbie downer to the developing world.

These days, I'd say a "prodigy" would be a titled master at the age of 10 or less ...

Shampoo

become titled before turning 18

2300 by age 12 has become the standard goal for predicting the world's elite.

There are of course lots of counter examples, actually in fact child prodigies failing to live up to expectations has become more the rule than the exception not just in chess but in lots of things. A lot of them flatten out before they hit 2600 and a few even quit chess entirely. I'm going to guess that heavy pushing from parents heavily predict these two things happening.

However this kid who has broken 2400 really seems something special. He's 12 years old, and the last two times he played Hikaru online he beat him. Hikaru knows about him and was definitely not holding back much.

https://www.chess.com/games/archive/legendisback1?gameOwner=other_game&gameType=live&gameTypeslive%5B%5D=blitz&opphikaru&timeSort=desc

There may be SOME chance he is cheating, but there's no evidence for it. Meanwhile he's humiliating GM after GM on the board.

I'd consider @VuHuyMinh2013 a prodigy. He is very good for a 10 year old

It might be different ratings for different ages, like 2300 for 12 but 1100 for 7 or something like that.

A rare few young children seem to have an ability to "intuitively" pick up the ideas underlying chess (or music, or mathematics) and rapidly achieve mastery with less coaching and study than others require to become average players. Those are the real prodigies.

Factors such as when the person learned the game, how much time they spent playing, when and how often they had the opportunity to play rated games, and other factors will partly determine what elo rating they achieve at what age--there is no standard measurement of particular rating at a particular age to qualify as a prodigy.

I am a ten year old and i have a rating of 1200. Here is what I think.

5 year old - 400

6 year old - 750

7 year old - 900

8 year old - 1200

9 year old - 1400

10 year old - 1750

11 year old - 2000

12 year old -2200( people have been a GM at this age

13 year old - 2300

14 year old - 2400

15 year old - 2550

16 year old - 2600

17 year old - 2650

18 year old - 2700

19 year old - 2750

20 year old - 2800

yup, thinks like a ten year old