I didn't take his objection to be as you rephrased it. I took his objection as the President "stretching" the AUMF beyond Congressional intent, when that stretching is used to justify surveillance with a domestic component.
At the same time, Specter seems just fine with "inherent authority," where any given (domestic/terrorist) surveillance would be measured in court by the test of "reasonableness" as articulated in the 4th amendment.
I'll have to read the transcript to see if he is adamant about requiring court order for surveillance in the first place. I think he's not adamant that way.
Okay...I have to admit that I was listening to Specter while playing with Chase...
BUT, I really got the impression that he seemed SHOCKED that this kind of surveillance was being used under the AUMF...I didn't catch whether he was referring to "domestic" use.
Please let me know what you find in the transcript...I think these guys like to talk "lawyerese" so that they will impress us...and I think they get too cute...like Graham does.