I live in London mate, I was making a distinction between a train that is operated by TFL and a train that is not. I also didn't call it a tube because I was referring more to the overground.

TFL Rail is the name for TFL's temporary services either side of what will be Crossrail. Once Crossrail opens to the public it will all become Crossrail / The Elizabeth Line even those services that don't actually go under London.

So TFL do operate trains, even ignoring London Overground which is in effect also TFL.

Anybody who is aware, that TFL operates trains too :)