I know. But their marketing department doesn't care.
Personally I think an "Alternate Training Site Site" Squad would be cooler.
We're lucky they weren't given a fake team name, like the Li'l Scrapple Squad.
I've never known the major league writers to bother following the vagaries of minor league rosters and rules, but I don't think you can really blame them this time. The information doesn't seem to be out there, and it's certainly not consistent. And at this point, "assigned to Lehigh Valley" in layman's terms means someone is physically in Lehigh Valley. No one cares which minor league roster they are on procedurally or developmentally.
The latest Phillies transactions say that Listi and Stott have been 'Invited to spring training," which is another amusing bit of semantics. If they ever get around to calling up Sanchez, presumably the official transaction will have him being recalled from Reading. Though Austin Davis, who presumably was on the Iron Pigs roster, shows as being "recalled from Phillies Alternate Training Site" (whereas Irvin, McLain and Garlick were recalled from the Pigs).