Non-negative random variables (ask about the definition) As I understand, a non negative random varialbe which takes non-negative values. But I've seen in that in many books the definition for non-negative random varialbles is a random variable $ X $ such that
$ \mathbb{P}\left(X<0\right)=0 $.
How come those two definitions equivalent? $ \mathbb{P}\left(X<0\right)=0 $ does not imply that $ X(\omega) \geq 0 $ for any $ \omega \in \varOmega $.
What is the acceptable definition between the two that  I mentioned?

You probably have heard about Murphy's law. Aside all the rhetoric and myths around it, the Murphy's law actually is quite important. An event can be possible or impossible. Probability is only defined over possible event. A possible event can be improbable, i.e., its probability can be zero, but it doesn't mean that it cannot occur.
But as you mentioned, it is customary to assign zero probability to impossible events. Even though it is ultimately a bad practice, it usually works. The good practice however, is to always make a clear distinction between impossible and improbable events.
