See my comment elsewhere in this thread where I think I pretty much exhaustively treated the subject.

And no, that particular version of 'split phase' is not +120V and -120V, this is not DC that we are talking about. Europe used to have a lot of that kind of split phase but we 'phased it out' to use a cheap pun.

So now the term single phase is used to indicate what you typically get delivered to your house and we don't further subdivide it. This is good because it means you can run much thinner wire due to reduced current. You do get a bigger whack if you accidentally end up touching the phase or if there is an internal short in an ungrounded device that exposes that phase to the outside (this should never happen). In North America 240 V center tapped off a single drop transformer is the norm for residential delivery.

I should have said two 120V legs offset 180°.

That's accurate. In NL until the 70's we had two levels of delivery, 127V (phase to ground, transformer in 'star' configuration) and 220V (phase to phase, transformer in 'delta' configuration). The latter is interesting because it doesn't actually have a neutral wire at all.

The former was close enough that you could get away with plugging in US or Japanese devices, but if there was anything synchronous in there you'd be out of luck (50 Hz vs 60 Hz). Then 220V became the standard and now it is all 240.

Interesting tidbit: there was so much inertia from older systems that needed upgrading in this that it took until 2004 until the last of it (the 127->220V change) was finally done.