In an application received by a broker, do we report the application date received by the broker or the date that it was received by the institution.

What date you report depends on the policy at your institution. The last bank that I worked for reported the date the borrower signed (not the date printed next to the loan officer's name by the LOS) for the application date on third party applications. Where I currently work, we report the date that our institution received the application in our underwriting department. Either of these are acceptable, though I prefer the current policy I am working under. Consistency in your reporting is what counts. Remember, the broker may have shopped the loan around for months to different lenders before it arrived at your institution.

This is where you need to be careful about overlapping regulations.
We use the same date for HMDA and RESPA.
You need to clarify what your bank's policy is and follow it so you don't trip yourself up with examiners.

We use the same date for HMDA and RESPA.

If you have a pre-approval program this procedure won't work.

Agreed.

It wouldn't be a RESPA application without the address anyway so your procedure would need to address a preapproval situation specifically.