Nope, the priestly years are not from Seder Olam Zuta.
That's quite annoying. I did track down which Wikipedia editor originally added the years and left a message with them requesting that they provide their source.
Then there would have to be two 'Bagoases' as well.
I'm not seeing a problem with a single Bagoas being active during the reigns of Darius II and Artaxerxes II, which correlates with the tenures of Joiada and Johanan.
Priests had chambers around the temple area with different functions.
A few of the commentaries note that the High Priest (i.e. Eliashib) could allocate such 'chambers' (which were outside the temple) to whomever they saw fit.
They don't say Johanan was high priest yet - just that he was old enough to have a prominent, responsible position in the temple.
See my more recent post about Johanan and Jehohanan. Ah... you did.
Don't follow [6 years for Darius].
The 2 discrepancies about those two differences of one year each in my chart were the result of rounding of months.
[Eerdman's Commentary on the Bible]
Because the list of High Priests and reference to Darius at Nehemiah 12:22 is probably a later interpolation, it is unsurprising that, as Garbe points out, the writer of the rest of the passage seems unfamiliar with more than one Artaxerxes.
It may be consistent with the fragments of information available, but it remains conjecture.
I'm very grateful for your eagle-eye on these matters. I'm going to incorporate as much detail as possible for the final version I eventually upload to my blog. I only see this as 'a puzzle to be solved', and not anything of monumental importance, and definitely not anything remotely mystical. I maintain that Nehemiah 12:22 refers to Darius III as a better fit though.