Cigarette lighter = Empower.
You need the Empower connection for FIRST or old NCW. You use a US/European adaptor for NNCW.
You have have to pull the cigarette lighter cover attachment off to get to the smaller attachment underneath for use on BA.
Even if you know that you are in NNCW, always worthwhile to have a Empower connection in reserve if you ever get upgraded to FIRST
Edited: Did not see that the Op is based in the US so a US adaptor is unnecessary. Also a joke that
British Airways to do not take British electrical plugs but US and European ones.