Going to go against the flow here and say don't bother with a NASA report, but of course it doesn't hurt. While you violated the letter of the law you didn't violate the spirit of the law. You were in two-way communication with approach the whole time. Additionally, approach can clear you to enter the Delta and clearance for an approach at a Delta is, in my mind, clearance to enter the Delta.

But the Granby LOI (15181) clarifies that for Class C 14 CFR 91.130(c)(1) is always applicable regardless of whether you're in communication with another ATC facility, and for Class D in your case 14 CFR 91.129(c)(1) is identical. In your specific instance, you would need to know what Approach and the Class D tower had for a Letter of Agreement and what the approach controller had coordinated with the Delta tower. But you could have only got this by asking, which you didn't.

From a safety aspect, approach is supposed to coordinate with the appropriate control tower for transit authorization when providing radar traffic advisory service to an aircraft that will enter another facility’s airspace. So you did nothing unsafe, which is what the NASA reports are primarily about.

But you are correct that the responsibility in getting/verifying that clearance to enter/transit an airspace remains with you as the pilot in command. So even if approach is coordinating you need to get confirmation from approach that they've coordinated your entry or contact the tower and establish 2-way communication.