Bernanke has regularly referred to "Core Inflation" when discussing inflationary pressures. You can see his use of it as recently as Feb 3, 2011 in his speech to the National Press Club:
http://www.federalreserve.gov/newsevents/speech/bernanke2011...The Fed's version of core inflation is the core PCE price index as you stated. The "core PCE price index [...] excludes the more volatile and seasonal food and energy prices" [1].
While you may reasonably take issue with my wording of "quite narrow" as I perhaps wasn't choosing my words carefully enough, core inflation/core PCE certainly isn't "the most general measure of inflation that is theoretically possible". I assume you weren't meaning to refer to "core PCE" - but never the less it is clear from federal reserve speeches that it is this measure that has been playing significantly into federal reserve monetary policy.
Core inflation makes sense in trying to smooth out seasonal or short term moves. However, the effects on global food and oil commodities that are traded in US dollars seem to far exceed any seasonal or short term influences. In that case discounting food and fuel influence on actual inflation rates seems unwise.
Bernanke seems to be acknowledging this. The WSJ reported on the 13th of March in an article titled "To Avert Criticism, Fed Avoids Saying 'Core'": Now, Mr. Bernanke is taking another tack in his public communications. In two days of testimony on Capitol Hill earlier this month, the Fed chairman never uttered the words "core inflation" while explaining the central bank's aims and policies. [2].
As a side note, as I've noted I'm in no way trying to put myself forward as an expert on these subjects. I get a significant amount of my financial news and analysis from the WSJ and Marketplace (public radio) (and Morgan coverage, which admittedly is not particularly hawkish here). I'm not above admitting that I could be making some serious mistakes, so if you'd like to suggest better sources of information I am all ears.
[1] https://secure.wikimedia.org/wikipedia/en/wiki/Personal_cons...
[2] http://online.wsj.com/article/SB1000142405274870489360457619...