Expected value of order statistics from uniform distribution

You cannot get Equation $2$ from Equation $1$, because they are not the same. You've mixed up your $r$ and $k$. Notice how in your proof of Equation $1$, you wrote

$$\operatorname{E}[X_{(k)}^r] = \int_{x=0}^1 x^{\color{red}{k}} f_{X_{(k)}}(x) \, dx.$$ This is clearly wrong, because you wrote an exponent of $r$ on the left hand side, but the exponent on the right hand side is $k$. If you do the computation correctly with $r$ instead, then you will get

$$\operatorname{E}[X_{(k)}^r] = \frac{n! \, \Gamma(r+k)}{(\color{blue}{k}-1)! \, \Gamma(n+\color{blue}{r}+1)}.$$ Then this equation is consistent with Equation $2$ from your original question.