Fourier transform of dirac comb with function: The scaling factor
Your first formula is wrong. If you use angular frequency ($\omega$), then multiplication in the time domain corresponds to $1/(2\pi)$ times convolution in the frequency domain, just as in the last formula in your question:

$$\mathcal{F}\{x(t)\cdot y(t)\}=\frac{1}{2\pi}X(j\omega)\star Y(j\omega)\tag{1}$$

If you use frequency $f=\omega/(2\pi)$, then you get

$$\mathcal{F}\{u(t)\cdot v(t)\}=U(f)\star V(f)\tag{2}$$

where I've used the definitions

$$X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\tag{3}$$

$$U(f)=\int_{-\infty}^{\infty}u(t)e^{-j2\pi f t}dt\tag{4}$$