Your institution needs to define its own application process.
What is an inquiry/prequalification? These are not reportable. If you deny them, they still aren't reportable applications under Reg C, but they do become applications under reg B and require AAN.
What is an application? "Application means an oral or written request for a covered loan that is
made in accordance with procedures used by a financial institution for the type of credit requested." That is for you to figure out. Many institutions use the TRID 6 pieces, but you can't treat them like a request for credit before this point if you want to use this definition. If you are sending NOIs and closing files as incomplete without an address, then you're going to be hard pressed arguing they aren't applications seeing as those are steps taken with an application.
What is a completed application? This is irrelevant for whether an application is reportable under Reg C, but you should know this as well for Reg B purposes.
I have seen commentary that states we should report on the LAR if we have enough information to complete even though we don't consider a complete application. I have also seen where we do not have to report if we considered an incomplete application.
Where have you seen that? Neither of those statements seem accurate or really make sense. If it's an application, it's reportable whether incomplete or not. Question is whether it's a prequal/inquiry or an application.