Logan Grimnar, the High King

[pigei]

The high king rules states that at the beginning of the turn you may choose one of the special rule listed.
Then it states that for the duration of that player turn Logan and his unit have the selected special rule.

Now, suppose that I play second after my enemy. In this case "that player" refers to the enemy.
Suppose that I choose for Logan either RELENTLESS or TANK HUNTER.
As stated the special rule will last only for the enemy player turn and will not be available during my following turn.

Is this interpretation correct? I know, for instance, that a similar thing happens for Nijal's tempest effects

In this case if I play second I might only choose Fearless or Preferred Enemy (which might have effects during the enemy turn)

[Hive Mind]

I believe that unless otherwise stated all uses of 'turn' refer to player turn, not game turn.

So no, your interpretation is not correct and you pick the SR for Grimnar at the beginning of your turn.

[pigei]

You are right I always swap the meaning of "turn".

Anyway this means that you can choose a special rule both at the beginning of your turn and at the beginning of the enemy turn.

[Lerra]

Right - you can pick a rule for both your turn and your opponent's turn. It's handy to be Fearless against shooting and then have Preferred Enemy during your turn when you assault.