I want to follow up here in case someone else has the same issue in the future -
I just spoke to both Diner's Club and Avis Customer Service.

For Avis free days, Avis actually "runs" your card for the regular cost of your rental (as if you didn't have the free coupon), then when you return the car, they deduct the coupon value for the free day, leaving you with just the taxes. So in this case (according to both Diners and Avis) you would be covered by DC for insurance.

When using any company's "free day" vouchers, the insurance on a "free" car all hinges on them running the rental cost ahead of time (setting up a regular paper contract) on your DC and then deducting it after the return.

So, I guess I will start using these "free day" coupons they send me every couple of months.

On a side note: As I've posted before, I can't say enough good things about using DC for car rentals. They are in fact the only card with primary coverage and I had to use them a couple years ago when my National car was totaled in an accident. They could not have been nicer and best of all my personal insurance company never had to be contacted. It is definitely worth the annual fee.