A B737 flight crew deviated from the clearance route when they misinterpreted their PDC.

Narrative

PDC clearance confusion ZZZ-BUF: PDC clearance ZZZ (SID) XXXXX J220 VALLO. Filed route on PDC and releaser KZZZ ZZZ (SID) XXXXX J220 BUF. I interpreted after VALLO cleared direct to Buffalo. After we passed VALLO ZNY queried our clearance; as they were expecting us to continue on J220 after VALLO. ZNY cleared us direct BUF and noted this has been a problem and also said there was no problem or conflict. I have reviewed company manual with this exact route and change as an example of what is expected. I did not ask for a full route clearance on the ground at ZZZ; as I assumed incorrectly; after VALLO we were cleared direct BUF. This obviously is still a problem with the logic of PDC clearance changes. It reinforces to request Clearance Delivery for full route clearances when I see any PDC route changes similar to this in the future. I think PDC changes should have a common end fix to pick up the original clearance. In my case; if the PDC change would have been ZZZ (SID) XXXXX J220 VALLO J220; it would have provided a common end point to the original clearance J220 BUF.