Does a character automatically fail one death save when they drop below 0 hit points?

Hey guys so I have a question regarding death saves and if anyone knows an official source that can determine this.

So according to my dms interpretation of the RAW, if an attack brings you below zero hit points, it not only knocks you unconscious but also inflicts an automatic failed death save from that same attack.

And I to this point was under the idea that the attack that brings you below 0hp as long as it doesn’t also do your max health in damage after the fact then you are simply unconscious. And then all damage received after that inflicts auto fail saves (crits inflicting 2)

Is my interpretation or my dms interpretation correct and is there an official source or creator that has spoken on the matter?