The clearest explanation of the Clausius-Mossotti (CM) relation I have ever come across is this paper by Aspnes. The correct definition of the dipole moment must always relate to the microscopic field acting on the individual lattice sites. It is this microscopic field which induces the dipole moments. The microscopic field is different from the apparent macroscopic externally applied field. The latter is the sum of the microscopic applied field and the volume averaged dipole field (related to the macroscopic polarisation field). This is exactly what your second source is saying with $$\mathbf{E}_{eff} = \mathbf{E} + \frac{\mathbf{P}}{3\epsilon_0}.$$ $\mathbf{E}_{eff}$ is the microscopic field acting on each dipole, written in terms of the macroscopically averaged electric field $\mathbf{E}$ and polarisation $\mathbf{P}$. The factor $\frac{1}{3}$ accompanying $\mathbf{P}$ arises due to the volume averaging. I don't know the details of Griffith's derivation, but his symbol $\mathbf{E}$ must denote this microscopic field also, or he has done something dodgy.

The rest of your confusion appears to stem from definitions and units. You are free to define the polarisability $\alpha$ so that $\mathbf{p} = \alpha \epsilon_0 \mathbf{E}$ or so that $\mathbf{p} = \alpha^{\prime} \mathbf{E}$. You convert from one to the other by $\alpha^{\prime} = \alpha \epsilon_0$, exactly as you convert between your corresponding CM expressions. The appearance of a $\frac{1}{4\pi}$ in place of $\epsilon_0$ is common when converting from SI units to other unit systems common in electromagnetism, where often $\epsilon_0 = 1$ by definition.

The clearest explanation of the Clausius-Mossotti (CM) relation I have ever come across is this paper, i.e. Local-field effects and effective-medium theory: a microscopic perspective, D. E. Aspnes, Am. J. Phys. 50, 704 (1982). (I apologise that I can only find a version that is behind a paywall.) The correct definition of the dipole moment must always relate to the microscopic field acting on the individual lattice sites. It is this microscopic field which induces the dipole moments. The microscopic field is different from the apparent macroscopic externally applied field. The latter is the sum of the microscopic applied field and the volume averaged dipole field (related to the macroscopic polarisation field). This is exactly what your second source is saying with $$\mathbf{E}_{eff} = \mathbf{E} + \frac{\mathbf{P}}{3\epsilon_0}.$$ $\mathbf{E}_{eff}$ is the microscopic field acting on each dipole, written in terms of the macroscopically averaged electric field $\mathbf{E}$ and polarisation $\mathbf{P}$. The factor $\frac{1}{3}$ accompanying $\mathbf{P}$ arises due to the volume averaging. I don't know the details of Griffith's derivation, but his symbol $\mathbf{E}$ must denote this microscopic field also, or he has done something dodgy.

The rest of your confusion appears to stem from definitions and units. You are free to define the polarisability $\alpha$ so that $\mathbf{p} = \alpha \epsilon_0 \mathbf{E}$ or so that $\mathbf{p} = \alpha^{\prime} \mathbf{E}$. You convert from one to the other by $\alpha^{\prime} = \alpha \epsilon_0$, exactly as you convert between your corresponding CM expressions. The appearance of a $\frac{1}{4\pi}$ in place of $\epsilon_0$ is common when converting from SI units to other unit systems common in electromagnetism, where often $\epsilon_0 = 1$ by definition.