"Sim" wrote
Now let's be nice to each other! I did not know the details Charles
kindly provided, but it does make sense. Interestingly, not only does
fourth rail (however wired) exist as far north as Harrow, but the last
time I looked there was quite a lot left further on, although some of
it was lying rather dismally in the four foot rather than perched on
insulators. It was never formally removed, in other words, although
doubtless disconnected.

On the more general point on whether Overground is part of National
Rail, I suggest that it is, behind the scenes, a remarkable compromise/
fudge/whatever.

Consider: Overground is a TfL operation, and the concession was
awarded to LOROL by TfL. Other posters have already explained the
differences between a National Rail franchise and a concession like
Overground. Overground is funded and branded by TfL and included in
its operations for all public purposes. Overground is, of course,
Oyster compatible along with DLR and trams (and buses too, yes).
Station specs (staffing, equipment, appearance, branding) have been
officially described as comparable with the Underground (although not
all the upgrades are done yet).

On the other hand, most Overground trains run (or will run) over
Network Rail infrastructure, and on some sections they share the line
with freight traffic (or the true Underground ). South of New Cross at
least, Overground will presumably share its tracks with scheduled
National Rail passenger trains, too. One section, though (Dalston
Junction west curve to New Cross Gate/New Cross inclusive) is TfL
owned/maintained infrastructure.

The rolling stock was specified by TfL and is leased by TfL, but is
included in the NR Rolling Stock Library as Class 378/x, being yet
further variations of the Bombardier Electrostar series (and thus
thankfully built in Derby!).

Overground is also being treated by the Office of Rail Regulation as
part of National Rail: its statistics are included in National Rail
Trends just as if it was another franchise. The ORR does not report
the figures from other TfL rail systems, any more than it includes
Tyne & Wear Metro.

And another poster has also rightly pointed out the existence of 25kV
in various places, which is why dual-voltage roilling stock is needed.
To add a little savour, parts of the 25kV NLL route (Camden Road
area?) have third rail as well as OHLE -- a rare combination, I would
suggest.

If I came across such a compromise system in a foreign capital, I
would be intrigued.

As it is, it's in London. Hooray!

That seems like a reasonable summary. Now, can someone explain succinctly
why the smaller 'National Rail' stations between Queens Park and Harrow &
Wealdstone, used by LU and LO, which were previously Silverlink-branded, are
now signed as LU (rather than LO) stations?