ASCAP and BMI are not publishers. They are PROs (performance-rights organizations).
The difference: Publishers are members of ASCAP. ASCAP collects royalties and distributes those royalties to Artist members and to Publisher members.
A third "significant" PRO usually included in this type of conversation is SESAC.
You should consider correcting this: "According to ASCAP (one of the big two publishers, with the other being B.M.I.)" to something like this:
"According to ASCAP (one of the three big performance-rights organizations, with the others being B.M.I. and SESAC)"