User blog comment:Deedlit11/Ordinal notations II: Up to the Bachmann-Howard ordinal/@comment-11227630-20170727144618/@comment-5529393-20170730073739

There's no difference in the end between the two. The difference in general is that if A is a subset of B, then A doesn't have to contain all of B, whereas of course this is not true with A = B. But C(a,b) is defined to contain all of b to start with, so if C(a,b) intersect Omega is contained in b, it must be exactly b. The Psi function in "Proof Theory of Reflection" uses =, but here again it would be the same if it used "subset of".

If you have any more questions about that OCF, feel free to ask.