One of my major gripes about slashdot is that it doesn't allow editing of one's comments.

PLEASE LET US EDIT!

This 8 move game disproves your claim:

      . .
      . .
A2
      X .
      . .
B1
      X .
      . O
A1
      X .
      X O
B2
      . O
      . O
A1
      . O
      X O
A2
      O O
      . O
A1
      . .
      X .
B2
      . O
      X .
A2
      X O
      X .
pass
pass

Btw, we're counting all games, not just games without mistakes.

You might want to learn how to play Go:-)

That's NOT how it's used in chess.
Even though the notion of chess position might include some game state information
  like side to move, castling rights, and en-passent capturability,
I've NEVER seen it include the set of all previous board states, as would be necessary
for applying the 3-fold repetition rule in the game's future.

And the number of legal chess positions is in between the number of legal 9x9 Go positions and the number of legal 10x10 Go positions...

2x2 is Go played on a 2x2 board.
where the 4 corners are the playable points.

A game can continue for dozens of moves.

You mean 3^(19*19), a MUCH bigger number.

Yes, I counted the number of possible board states, which I call positions.

You propose to instead use the word position to describe the complete game state including set of previously visited board positions.

I say that's really confusing.

The 386356909593 games do account for superko.
That is exactly the number of simple (meaning not visiting the same point twice) paths starting
from the empty node in the graph in Figure 4 on page 5 of our paper http://tromp.github.io/go/gost...

Without superko, the number of 2x2 games would simply be infinite...

The notion of legal position namely "every group of connected stones of the same color having an empty point adjacent to it" is NOT dependent on any particular choice of go rules.

Only legality of moves is...

Author here.

A single 2x2 game can visit as many as 48 of the 57 legal 2x2 positions, with many dozens of passes in between moves, and obviously many captures.

This page

http://tromp.github.io/java/go...

on solving 2x2 go with various search methods may be helpful. I've lost track of my original 2x2 game counting code but suspect it was a close relative of this code.

The only position not having a color-swapped sibling is the empty position...

"to a doggy day care"

I originally read that as "dodgy day care" ...

-John