TPE has never been "InterCity".
Until approx 1991, they did run decent long trains with loco hauled stock, it wasn't operated by InterCity but I guess it was not far off in terms of the length of train and calling pattern. But since then it's been downgraded into a short formed DMU operation which certainly doesn't meet any of the criteria in by opinion.
I would argue that the Scotland services (ie, Glasgow and Edinburgh-Manchester Airport) could be classed as "intercity" operations. They have a limited calling pattern, a high level of service despite the DMU operation and, IMO, warrant intercity status. Given the history as well, I believe it's appropriate. However, I'd agree that East of Manchester services aren't really intercity on their particular stretch of line; they call at most if not all stations and really are more "inter-urban", same goes for services to Blackpool. Important services, of course, but not "intercity" as such.
The thing is, we could sit here all day debating what is and isn't "intercity" and still not come to a conclusion. Of course, we've pretty much agreed elsewhere that "Intercity" can't simply mean between two cities; that would mean that several tube lines are intercity trains. Most people would agree that Virgin and Cross country are intercity operations, whereas Northern provides a local service. But some, like the TPEs to Scotland, will always cause some debate.
