Oh, and the Circle line directions should be Clockwise and
Anti-Clockwise.
This is not an opinion - this is a fact, proven by science.

What is this "fact" proven by science? Personally, I find it much
quicker to relate Inner Rail and Outer Rail to a direction than
Clockwise and Anti-Clockwise, which I find unnecessarily indirect. (I
have to remember first which way analogue clocks move, which to a
mathematician is the "wrong" or negative direction. Why people still
want to tell the time by looking at the angles of two sticks is a
mystery to me.)

I find Inner Rail and Outer Rail easiest too. As you head along the
District, the Inner Rail goes "inner" the City and the Outer Rail goes
"outer" it. :-D This just goes to prove that the world revolves around
the District Line.

Perhaps (and this is not an entirely serious suggestion) we just
need to pick a pair of names which don't have any specific
geographical connotation and use those consistently along the
whole line: Up and Down come close, are nicely traditional, but
can't really be applied inside London; i suggest Ana and Kata,
these being the traditional extra directions in maths.

I think your traditions are rather younger than mine.

Alternatively you could translate the Greek words into English:
Ana=Up, Kata=Down. Then an Up District train would be headed for
Mansion House, Tower Hill, Whitechapel, Plaistow, Barking, or
Upminster. It makes sense. You know it does. The easiest place to get
an "Up" direction is from the stretch between Wimbledon and East
Putney.