Following up on a comment from Millard Humphreys, I would agree that although some of the specific words tend to change, the idea of a supply chain's reach extending from "the supplier's suppliers to the customer's customers" seems to be a pretty universally accepted premise.
I also like the succinct "from dirt to dirt" definition.
What all these terms and phrases have in common, of course, is the insistence that companies no longer can make it on their own (not that they necessarily ever could to begin with), especially not in a "flat" world heavily reliant on global sourcing to satisfy the demands of customers who may learn of your products via the web rather than in the local newspaper, as it did in days of old. As technology makes it easier to communicate, it also accelerates the pace of commerce -- and companies can no longer get by on simply having the highest quality products. If your distribution and logistics networks are hopelessly clogged, for instance (which represents the "deliver" piece of the SCOR definition), then you need to address that problem area, which means looking outside the four walls of your plant floor to figure out where the bottlenecks are.
Since many companies are no longer staffed to address their logistics needs directly, the role of the third-party logistics provider (3PL) has become increasingly important in recent years, with something like 80% of all North American companies outsourcing at least part of their transportation tasks or processes to a 3PL.
Here's a quick "how do you do it?" list of suggestions for companies who know they have supply chain problems but aren't sure if going the 3PL route is the right answer.