How do different formulas for calculating the mean anomaly of an elliptic orbit correlate?
Kepler's Third Law including the constants of proportionality is:
$$GM T^2 = 4\pi^2 a^3$$
Substituting $\mu = GM$, this can be rearranged to give:
$$\frac{2\pi}{T} = \sqrt{\frac{\mu}{a^3}}$$
Which lets you rewrite your first formula as your second one and vice-versa.
Note that in the first formula the reference epoch $t_0$ is assumed to be the time of periapsis, which corresponds to the case $M_0 = 0$. Your definition of $\Delta t$ should also read $\Delta t = t - t_0$, i.e. time since the reference epoch.