What is the correct solution for Fourier transform of unit step signal?
As Matt said, the second and third definition are the same except for the part with impulse. The impulse ($\pi\delta(\omega)$) account for the DC value of $u[n]$. Without that term (i.e., the second definition) it is actually the FT of $v[n]=\frac{1}{2}\operatorname{sgn}[n]$. We have $u[n]=v[n]+\frac{1}{2}$. And hence the FT of $u[n]$ has the additional term to account for the addition of $\frac{1}{2}$. Also, the discrete time FT (or DTFT) of $u[n]$ is properly written as $U(e^{j\omega})$.

The first definition, $U(j\omega)$ is the "continuous time" FT (or CTFT) of $u(t)$ (not $u[n]$) and hence different from the other two definitions.