There are few notes online stating that an all-pass filter is a maximum phase filter (e.g., Link). The core of the claim is that an all-pass filter is a maximum phase filter since its zeros are outside the unit circle, which is the definition of a maximum phase filter.
However, according to the MIT lecture note, a maximum phase filter is stable and anti-causal while its inverse is also stable and anti-causal. As we all know, for an anti-causal filter and its inverse to be stable, its poles must also be outside the unit circle, so its ROC includes the unit circle. The same claim can also be found in other notes (definition D. 3. 6.)
But going back to the definition of an all-pass filter, its transfer function can be expressed as:
Assuming $|a|<1$, its pole is inside the unit circle, but its zero is outside the unit circle. However, since its pole is inside the unit circle, the filter cannot be stable if this is an anti-causal filter. If this is not an anti-causal filter, then it is difficult to see it as a maximum phase filter.
Can anyone point out what I am missing here?