Is the Signal-to-Noise ratio arbitrary?
At first I wondered what to do with your question, because there was a lot of misunderstanding in it, but finally, let's just answer the core question:


  It looks to me that the Signal-to-Noise ratio (SNR or S/N) is completely arbitrary and it's set by the user according to his preferences.


No. That's wrong.

The signal power reaching a receiver is a product of the signal power transmitted by the sender, the path(s) between that sender and the receiver, the receiver properties and so on. Most importantly, it's the power of what you consider to be a part of your signal, based on your mathematical signal model, reaching your receiver.

The noise power, basically, is everything that you don't label as signal.

So, there's absolutely nothing a developer chooses. A dev will try to optimize the SNR, but it's an effect, not a design freedom.