The term “old ways” refers to their cultural practices as well as their magical ability. The Psijic monks practice non-magika magic that was used in the Dawn era, based on its description in the book “On Atreum” it’s similar to the Th’um and Sword Singing.

In regards to their philosophical belief, they are what the Altmer fail to be in my opinion. The Psijics view themselves in two different ways from what I interpret.

On one hand they act as agents for Anu by preserving his Dominance over the forces of Change. To them the Aedra are merely spirits like them who are shaped by the greater forces, and it’s their duty to “protect” these spirits by understanding how to “guide” Padmoay’s chaotic influence.

On the other hand they are also dedicated to emulating their ancestors and practice the teachings of Auriel that Mer society in Tamriel has abandoned. As far as I’m concerned the knowledge on how to ascend like Auriel has been kept secret by the Psijics from the corrupt Mer who believe they are entitled to the “Old ways”.