A variety of joseki that bots appear to like, but I rarely see humans play (except the first one). First two are more common in 11x11.

• https://www.trmph.com/hex/board#11,i8i9h9
• https://www.trmph.com/hex/board#11,d8d7
• https://www.trmph.com/hex/board#11,h7h6
• https://www.trmph.com/hex/board#11,h7g8h8h6
• https://www.trmph.com/hex/board#11,h7g8h8h9g9h6
• https://www.trmph.com/hex/board#11,e7d10b9
• https://www.trmph.com/hex/board#11,e7d10b9b10c10c9d9
• https://www.trmph.com/hex/board#11,b8d10c10b9d8d9c9

Bot opening motifs on 13x13. Top players will recognize most of these:

• c2 and b5 for black, 4-4 obtuse corner for white: https://www.trmph.com/hex/board#13,c2d10j9j4b5 (half of games by top players seem to start this way :D )
• If black opens with a7, then b8 is a strong followup (likewise with a9 and b10, etc.): https://www.trmph.com/hex/board#13,a7e9b8
• A good response to a10 (and the weaker a11) is b12. White often follows up later with another move on the left edge, either e4 or d11: https://www.trmph.com/hex/board#13,a10b12j9e4

Nice summary!

Some of the positions above I would say are not joseki, but just a move that happens to be adjacent because of the small board.

Yeah its a great mistery how to play C2 opening. Recently people learned 4-4 from bots but I'm not fully convinced to it. But c2-b5 to claim both long diagonals also looks far too naive. I personally prefer 5.g7 or another central move, but both won and lost some games that way…