See in the first case, there is an equality also, between $n$ And $N$, but in the second case, there is a strict inequality between $x$ and $N$.
So what I want to ask is that if we remove the equality in the first case and add an equality in the second case, do the definitions still hold the same meaning?
And to prove the equivalence of both the definitions we would have to prove "if and only if" condition, that is, first definition implies the modified one, vice versa. So that's what I am looking for.
For fixed $\epsilon>0$, if such an $N(\epsilon)$ exists in the functional definition, then define $N':=N+1$, so that the result holds for $x\geq N'$. In other words, equality is immaterial to the definition.