Why were we using $I=4I_0 \cos^2\left(\frac{\phi}{2}\right)$ instead of the one on the formula sheet and in the book?

The formula $I=4I_0 \cos^2\left(\frac{\phi}{2}\right)$ is the more general formula for a two-slit interference pattern: it gives the intensity as a function of phase shift $\phi$ (this is also in your text, equation 37-13). The expression $I=4I_0 \cos^2\left(\frac{2d\sin\theta}{\lambda}\right)$ just substitues $\phi=\frac{2\pi}{\lambda}d \sin\theta$ for the phase shift.