User blog comment:Nayuta Ito/faketest/e0/@comment-30754445-20180805015451/@comment-30754445-20180807034655

As a side note:

Just because the cofinality of the ordinals inside the ψ are different, does not necessarily mean that the final collapsed countable ordinal is different.

For example, the ordinal  ψ(Ω2) is sometimes written as ψ(ψ2(Ω2))... yet the cofinality of of Ω2 is Ω2, while the cofinality of ψ2(Ω2)=ζΩ+1 is ω.

So the difference between Rathjen's notation and Deedlit's might be nothing more than a cosmetic variation. For all we know, ψχ ε(M+1,0) (0) and ψ(εM+1) might even be the same ordinal in Deedlit's notation.