I'm pretty sure this has been covered somewhere, but lemme lay down how I think this goes:

Start of round:

Player with initiative has opportunity to play cards. Plays nothing.

Second player has opportunity to play cards. Plays Take Initiative.

At this point, the first player can't then counter with his own take initiative card because his opportunity to play cards has passed.

is this right? Both players couldn't play Take Initiative?

You are correct.

The person who initially had initiative has no chance to play it.

The first player already passed his start of round trigger.