Getting the wanted light from the Den

First off, you'll need to know whether you want a common nest, rare nest, or promoted nest.

Despite that some people and sites call those Dens, given the actual data in-game is called "nestholes", and the fact that it is important to differentiate between the two, for the sake of this tutorial, let's split them into nests and dens

Dens: the physical structure in-game you interact with. Enables you to access the pool from either the common nest, rare nest and promoted nest using a Wishing Piece.
Nests: the collective pool of Pokémon in a nest. Nests are not able to interchange with one another.
For example, if you have the purple light (rare nest), you cannot access the pool of the red light (either common nest or promoted nest)
Likewise, if your Wishing Pieced den gives you promoted nest, even though it is red light, you cannot gain access to common nest.

Now that the terminology is out of the way....
To be clear, this guide needs Wishing Pieces. From this point on, you'll gonna need to use Wishing Piece on dens...

Wanted light: Rare Nest
To get purple light (rare nest) from dens, we will use a game exploit.
1. Set text speed to Slow. (Note: this also works on Medium, but only if the user is fast enough)
2. Save the game x2 before the intended Den. (To retain the text speed)
3. Now throw a Wishing Piece into the den. Do not hold down the A button.
4. The light will appear faster than the "Saving" message.
if the light is not purple, go to Home of the Switch as fast as you can (before the saving message completes), and immediately close the game.

This prevents it from Saving, meaning you have not used the Wishing Piece (how economical!), and you can try again.

What if you weren't fast enough? Either:
1. Complete the raid
2. Throw a Wishing Piece into a different den. There can only be one Wishing Piece active at a time, and this will result in the previous one despawning.
 

Wanted light: Promoted/Event Nest or Common Nest
While you can use the previous method, and focus on red light instead of purple light,
there is a chance you'll end up on the nest you do not want. (You can tell which nest you have access to, based on what is the sprite of the raid).
As such, an alternative method was born.

This method requires you to have a lot of watts, and bought a considerable amount of Wishing Pieces.
1. Choose 2 dens that are near each other. Let's call them A and B
2. Throw Wishing Piece into A.
3. See what light you get from A, and if the species belong to the nest you want. If not...
4. Throw Wishing Piece into B. This will despawn the piece at A.
5. See what light you get from B, and if the species belong to the nest you want. If not...
6. Loop back to step 2.

Basically, you'll rotate between A and B until you get the nest you want.