I don’t recall this ever being brought up during any of the quests (I think). So what is our rank or position in the Psijic Order? Are we more of an honorary member of the order like Sotha Sil? Or are we fully in the Psijic Order?
Once you finish all the Augur of the Obscure quests, the achievement you're given is called "Psijic Sage." Assuming this is a rank, that could be it.
We also see Psijic Sages in the quest "Buried Memories" following other players, so we don't see multiple Iachesis NPCs running around. Source
Even though "Sage" sounds like more of a broad term to describe anyone in the Order, the other achievement titles surrounding the Augur quests would suggest it is a rank of sorts.
Lore-wise, I personally believe it makes a lot more sense for our character to just be a helping hand who was given honorary privileges for the help we gave. Unless they plan on ending ESO with a Dragon Break to canonize everything we did, while also washing it all away. More info on my personal theory below.
The Staff of Towers, the thing all the Augur quests built up to, was used to cause a dragon break. Even Josajeh wanted to use it to bring her parents back, leading to her getting banished. At the end of the questline the Staff is seen in the Vault of Moawita, fully intact. I think this was very deliberate by ZOS, it gives them a fitting end to the story of ESO whenever they're ready for that to happen. Or another questline surrounding it getting stolen or something. The Staff of Towers is an extremely dangerous artifact and the fact it's being kept in a vault in Artaeum seems pretty obvious as setting up a conflict that could lead to the Dragon Break to canonize all ESO characters' actions. This also would make it canon for all of the characters to have joined the Psijics, since in the lore there had only been 17 new members of the Order from like 2E 200 to 3E 432. While ESO takes place in 2E 582.