and the Boston MBTA transit
blue line switches near Logan Airport.

I think that changeover on the T happens when the train is berthed at
the station, whereas trains on the New Haven do it on the fly.

It's been a while since I've taken the train from the airport but I'm
pretty sure it's on the fly.

For added confusion, Penn Station in New York has both third rail and
OHLE, on different services but sometimes on the same tracks.

Yes, but 3rd rail at Penn is all overriding, whereas Metro-North has
only underriding.

There is a direct connection from the New Haven line into Penn via the
Harlem River Branch, which diverges just west of New Rochelle station.
Amtrak trains are now the only trains to run over that line, though the
MTA would eventually like to see New Haven trains running along it.
One of the potential difficulties for this prospect is that M-2 and M-8
EMU trains have only underriding shoes.

I don't see why that's a problem, since the OHLE runs into Penn
Station and beyond. There's an occasional MTA football special from
New Haven that runs through Penn Station to Secacus for the
Meadowlands stadium. Or are you saying the shoes would do bad things
with the LIRR's third rail?

There's also the Empire Connection, the former freight-only line down
the west side of Manhattan that allows Amtrak trains from Albany to
come into Penn Station. It's mostly unelectrified but there's a
little bit of third rail at the end that lets the trains run into Penn
Station.