You are probably going to get a lot of differing opinions on this, but like others have said, what you consider the application date is something your company needs to decide. I then suggest putting it in you published HMDA policy and being consistent on it. A few notes:
You mention the 6 pieces related to a TRID application. Many companies successfully use the TRID application date as the HMDA application date, but the two are not really the same. A HMDA application is defined as "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." Most people say that the "in accordance with the procedures used by a financial institution..." just relates to the application methods you accept (mail, internet, FTF, etc.) and is not a free license to put qualifiers on having an application. So you do not necessarily need the 6 pieces to have a HMDA application and it is probably possible to have a TRID application without a HMDA application.
I also don't really agree that the credit pull date is always the HMDA application date (though it usually is and is probably the safest thing to use). If a borrower calls and just wants to discuss rates or types of loans, then I don't necessarily think it is a HMDA application. Having an application is one permissible purpose for pulling credit (though application is not really defined in that purpose), but others are if you have written permission or if you you have a "legitimate business need for it in connection with a business transaction that is initiated by the consumer". In either of those cases, it is possible they haven't made an "oral or written request for a covered loan". Also, "application" and "business transaction" are not really defined in FCRA that I know of, which complicates linking the FCRA application to a HMDA (or any other rule) application.
The HMDA rule allows you to "report the date it received the application, as defined under § 1003.2(b), or the date shown on the application form." It would have been nice if they had specified which "date" they are referring to on the form, but I think you have a pretty good argument if you always use the earliest signature date on the form for the application date.
It might not hurt to email a question to the CFPB on it. There is a 98% chance you will get no response or a canned response, but if someone later doesn't like your procedure you can at least say you tried to get some official guidance but were ignored.