A direct extract from my book which states.(I have attached a photo as well)
"A point P(Xo,Yo) on the curve of a function y=f(x) is a point of inflection if f"(Xo)=0 and either f"(x) changes its sign at X=Xo, or [third derivative] i.e. f"'(Xo)≠0."
We can find pretty many examples for point of inflection where f"(Xo)=0 and f"(X) changes its sign at X=Xo. Like Y=X^3 at X=0. f"(X) is negative for (-∞,0) and positive for (0, +∞). So 0 is a point of inflection.
But i am really having hard time finding an example for second condition
By "second condition", do you mean
If $f''(x_{0})=0$ and $f'''(x_{0})\neq0$?
It implies the first condition. If $f''(x_{0})=0$ and $f'''(x_{0})\neq0$ then $f''(x)$ has different signs on either side of $x_{0}$. This is because $f'''(x_{0})$ is the slope of the tangent line to $f''(x_{0})$ at $x_{0}$.
The example you gave can still be used: $f(x)=x^{3}$, therefore $f''(x)=6x$ and $f'''(x)=6$. At $x_{0}=0$, $f''(0)=0$ and $f'''(0)=6$, and $0$ is an inflection point.