I agree with what you are saying. I'm not sure it was explained to the borrower by using the words "paying points" but from what I've gathered so far, the lender charged the fee because of the cash out loan purpose but also stated the borrower would have had a higher rate if the 0.625 wasn't assessed.
To make sure I'm understanding the difference and relay the information to the lending staff, because this scenario involved a rate option for the borrower and the borrower paid a fee to have a lower rate, it's a discount point. Had we charged a fee because the borrower had a lower credit score or higher loan to value (anything not rate related), it would be a loan level price adjustment on the TRID disclosure?