Why is boiling water the second time more quiet than boiling it the first time?

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.

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).