Phase spectrum: $\mathrm{arg}\mathcal{F}(\omega)$ or $\mathrm{-arg}\mathcal{F}(\omega)$?
Both definitions are common. However, if you talk about the phase, then what is usually meant is the (frequency dependent) phase angle $\phi(\omega)$ of the Fourier transform $F(\omega)$:

$$F(\omega)=|F(\omega)|e^{j\phi(\omega)}\tag{1}$$

The negative phase $\beta(\omega)=-\phi(\omega)$ is often called the phase shift or phase lag. It is commonly used together with the attenuation $\alpha(\omega)$, which is the negative natural logarithm of $|F(\omega)|$:

$$\alpha(\omega)=-\log (|F(\omega)|)\tag{2}$$

$$F(\omega)=e^{-(\alpha(\omega)+j\beta(\omega))}\tag{3}$$