Thank you for your answer, you helped my a lot.
To answer your question -> The Serial Monitor shows in all three cases the string I wanted to send (which was also a reason why I coulden't understand the different results)
And no I'm still checking the box if it is a problem on this side (which seems likely after your explanation)
If 2.1 is not a correct use of the function are you able to explaine the example from the documentary? (see: Serial.write() - Arduino Reference) There they write function is used the way I was trying with 2.1
Thanks for your help