Traditional ICF or Total ICF: Which is better?

This has been a long-time discussion among apeirologists: what is the best version of the Ordinal Collapsing Function: the one where the inaccessible variant ψ_I covers all ordinals of initial cardinality, or the variant that covers only the fixed points ω_α = α?

The difference between the two is that while ψ_Ω(ψ_I(0)) with a traditional ICF is ψ₀(Λ), and ψ_Ω(ψ_I(0)) with a total ICF is ω^ω or ω (depending on what you conisder Ω₀).

The traditional ICF came first because it is easier to define: it was first defined in the 1990s by Wilfried Buchholz. However the Total ICF took longer, with the creation of the χ function (which is identical to the Total ICF's ψ_I).

Both have their advantages, Traditional ICF keeps it's subscripted Ω's still in use, whereas with Total ICF's they're just sugar for the ψ_I function. In the end, it doesn't matter if you ask me.

Fun fact: the catching point of the Total ICF and the Traditional ICF is at ψ_Ω(ψ_I(I^ω))