Why is boiling water the second time more quiet than boiling it the first time? First of all: This is a different question than Why is boiling water loud, then quiet?, although the answers might be similar.
When I wake up, I boil some water for a cup of tea. It happens quite often that I do something else for 10-15 minutes and forget the water, so I boil it again. I noticed that boiling the water the first time is much louder than boiling it the second time.
Why is this the case? Did the air dissolved in the water disappear and therefore it became more quiet? How long would it take until the water has again air (I'm sorry, I know this is linguistically wrong, but I don't know how to write it correct). Is there any simple way I could get air again into the water to check if it's only the air?
 A: Water could have small pockets of air dissolved inside it, and given enough time they will rise to the surface on their own. 
Now when you boil water, what happens is that the vapour pressure at the water surface is equal to the atmospheric pressure, and liquid water gets turned into water vapour. This process does help the trapped pockets of air reach the surface more quickly, since the water molecules are moving around a lot during boiling. So after one boil, there is less water in your kettle, but there is also a lot less air dissolved in the water. 
As the question you linked to pointed out, some of the sound comes from the dissolved air bubbles hitting the surface, and with less air that is less sound.
Also quoted is the vapour bubbles that are produced at the bottom of the kettle (where the heating is). A vapour bubble is basically a bubble of water vapour, as the name suggests. The pressure of the water vapour is higher than the pressure of the water around it, so it rises upward to the surface. This produces a sound (quoting the paper linked in the question) of about $35 - 60$ $KHz$. 
But the second time you boil the water, the water isn't at room temperature. It's quite a bit higher. Therefore I suspect that less vapour bubbles are produced near the bottom (since the pressure of the surrounding water is higher than at room temperature, because it's hotter) and thus overall less sound. 
A: Warm and especially hot water does not absorb oxygen. This is the reason why ships, submarines and oil platforms rust much faster in the northern seas than in the tropics. 
So, as you yourself said, the reason is the same. Less oxygen is dissolved in the water and it makes less bubbles to pop and make noise. (And I think they pop and do not rise to the surface like the vapor bubbles that appear when the water finally boils at 100 °C).
A: the "roaring" noise produced as the kettle approaches the boiling point is caused by vapor bubbles which are suddenly initiated at pits and crevices in the walls of the vessel, and then suddenly collapse. those pits contain air and they remain active as vapor bubble nucleation sites as long as some air remains in them.
the second time you boil the pot, most of the nucleation sites have been exhausted of air and are inactive, so the roaring phase does not occur.
A: I don't know whether that's the reason.
Put the water into a large jar with a tightfitting lid. Shake the jar a lot so the water splashes around. That will help more air get into the water.
I can't think right offhand of anything else this treatment would do to the water which would be a confounding variable. But if you boil the water after that and it's still quiet, then probably air in the water is not what's doing it.
If it isn't air in the water, you might perhaps try condensing the steam, and see whether the water which steams away is louder than the water which stays behind. I wouldn't expect that to happen, but you'd be looking for something unexpected.
Air in the water is the only thing any of us have thought of so far, so we assume that's what did it and we assume that adding air to the water will make it boil louder.
If you get the chance, measure how loud it is. Your subjective comparison is probably right, but having it measured makes it more certain.
