"I know that report does not exist with the information I mentioned above."
Of course it does, and it a number of perspectives. Just one example is the Reports menu > Vendor Reports, the 1099 reports. You can run them and Customize them to ignore Thresholds, and for All Vendors, not 1099-Misc only.
"My problem seems to be occurring from a refund I received. Since it was for a job (I post everything to a job) I created a “Bill Credit”, posted it to an expense account, not an item because it is an Overhead Job."
What matters here is that the Job Reporting relies on Items. Your "overhead" if that still has a Job Name, you should always be using items because that is what the Job Reports are showing. You don't want one Huge Value listed as "no Item" because that doesn't tell you anything about these costs, when you run Job-based reports, such as Job Profit Summary, Detail or P&L by Job.
"I then made a deposit using the vendor’s name in the “Received From” column and used the accounts payable account in the “From Account” column."
Yes, this is how you link the Refund to the credit; if you didn't need job tracking, you could use Deposit directly with no name at all. Needing two names (vendor and Job) = needing AP credit.
"I then went to the “Pay Bills” window and selected the bill representing the refund amount. Then I applied the credit by selecting “Set Credits”. The vendor account now shows the credit"
It should not show a credit balance, though. Once you enter the refund, it offsets the credit balance. Then, you Applied them, that status is no longer Open, but the math won't change because you linked them.
"but not under payments."
Underpayments? Credits show, the same as Bill Charges, from the Job Tracking that you did. That's why you do them. For instance, Job Profitability Detail for that one job name = the vendor credit will be a negative entry. Run Unbilled Costs by Job, change the Billable filter to Any. You see it here, as well.
"So, if I create a transaction report"
Transactions are not the Job data; the Details are the job data. That's why you have to be very careful about the Perspective of the report you start with. Example: Reports menu > Banking reports, Check Detail. Make sure to filter on Payee name = Vendors, to see job details here. Otherwise, banking reports with Vendor Name (which is Source Name) doesn't show Job (target) names from the Details, because Banking reports are the Check perspective; neither the expense or items tab details perspective.
"with all the different types of payments, this will not show on the report. The only way it shows is if I include credits which is not correct."
Yes, Credits will reduce the amounts you paid, and that is proper.
Remember that using the 1099-Misc reports will not include CC amounts, because you don't report these payment types for purposes of 1099-Misc reporting.
So, for instance: Job Cost Detail by Vendor and Job, or by Job and Vendor, shows Everything you asked.
