This is what I think:

Application is made to Broker and before the loan is approved or denied by the Investor, the customer withdrawals - Broker reports

Application is made to Broker and approved by Investor and then the customer withdrawals - Investor reports

Is this correct?

Application is made to Broker and before the loan is approved or denied by the Investor, the customer withdrawals - Broker reports

If the application has been submitted to the investor and it's withdrawn before they made a decision then IMO the investor still had an application that was withdrawn by the applicant through the broker and it would be reported by the investor as a withdrawn application.

Application is made to Broker and approved by Investor and then the customer withdrawals - Investor reports

Yes, but not as a withdrawal. If would be an approved but not accepted.

In our case, the applications would normally be withdrawn before anything is submitted to the investors. In rare cases, information is sent to the investor and before the investor makes a decision, the customer withdraws - In this case, I was under the impression we (the broker) would need to report this loan since the investor did not make a credit decision. So much emphasis is placed on whoever makes the credit decision reports the loan, it's hard to distinguish in this situation because technically nobody has made a credit decision at all. That's why I feel like the responsibility would default back to the broker. Any other thoughts on this?

You need to look at who has the ability to make the credit decision too.
If all the investor reports are loans they approve and close, their LAR is going to be very skewed.
HMDA wants the reporter to report the action taken on every application received, whether it's approved, denied, withdrawn or approved but not accepted.
Otherwise, you are not getting a clear picture of the reporter's actions.

If the application has not been submitted to the investor then the investor has no application to report on.

Thank you for your responses. I consulted with our contacts at each investor and it turns out, one of them will report every loan they receive an application on (whether or not a decision has been made) and the other two will not report unless they have already approved or denied the loan.