Why is recording from the sound card of the laptop not working properly?

The first note sounds crackling on the recording, and the rest of the notes are a squeaking fever of a mouse you barely hear…

I have tried so far with the preinstalled voice recorder of Windows 10 and with the downloaded No23 recorder 2.1, both the same result…

Can it be that you have not chosen the right device for the recording?

Maybe the microphone is broken - vll. Times with a USB-Mic test. Generally, however, you do not get anything reasonable with onboard sound cards. Since the quality is always bad.

Remove the sound card driver from Device Manager. The next time you restart, the driver will reinstall. As a rule, the problem is off the table. Further help can be found under Troubleshooting under Settings / Sound / Input.

So in Device Manager is: NVIDIA Virtual Audio Device (Wave Extensible) (WDM).

I think that would have to be the sound card. The laptop was not cheap (four-digit amount), and that's an onboard sound card, I dare to doubt…

In laptops, there are usually no expansion slots for a soundcard. So an onboard sound card is most likely. But that is not a bad quality feature, at least does not explain your problems. Even with an onbard card you can play acceptable sound.
Go to the manufacturer page of your laptop and download and install the latest sound driver. Alternatively follow my recommendation from the top.