VAG has made it unnecessarily complicated, and it's their own doing, but it's no wonder the dealerships are confused too.
The PD motors (2004-2005.5 TDIs in N. America) require 5w40 oil certified 505.01. It says so right in the owner's manual, and probably on the oil cap as well. The confusion started when VW later (2006 or after) began certifying many 5w30 oils as 505.01. It still doesn't change that your PD motor requires the 5w40 flavor. Again, says so in your owner's manual. Adding to the confusion, VW's next diesel oil certification requirement was 507.00, which VW eventually started also saying was indeed backward compatible with 505.01. But 507.00 only comes in 5w30, so again, it is not suitable for a PD motor that requires 505.01 in the 5w40 weight. And I am not aware of any VW car sold in North America that requires 505.01 that allows 5w30 weight, so that adds to the confusion. But your car still requires 5w40.
Bottom line is 5w30 eats cams in PD motors. Don't use it or let the dealership put it in your aunt's TDI with the PD motor. I wouldn't bother arguing with the dealership about whether it is the correct oil or not. It is futile. They feel it is. VWoA probably agrees. But 5w30 will kill the cam. There is even a separate VAG part number for the 505.01 in 5w40: G-052-167-M2. It's a Castrol. Just have your aunt tell the dealership that is what she wants in her car and don't argue about the 5w30 505.01 or 507.00 oils. You have to spoon feed the VW dealerships... you will probably teach the parts department something new when you give them the part number.
That's if she insists on having the car serviced at the dealership. That Castrol 5w40 product the VW dealerships use is fine, but there are many other excellent 5w40 oils certified 505.01 that are readily available including Pentosin, LubroMoly, etc. Also, many TDI owners with the PD motor over at tdiclub.com have found great luck with some 5w40 diesel oils not certified by VW at all, including Mobil 1 Turbo Diesel Truck, Shell Rotella T6, Mobil 1 ESP Formula M, Schaeffer 9000. Those would all be much much better than using a certified 505.01 that is 5w30. The key is 5w40.