Like I said, I wasn't 100% sure, it has been well over two years since I've had a hero.
I'll reattack bat though with some clarity as I can see your perspective... I said every 3rd round you can get two attacks... What I have seen across multiple champions is this.
I base this on initiating combat. Once you are in combat we can't see timers so things get do not align perfectly.
Bat
Auto combat round 1
Bat
Auto combat round 2
Bat
Bat
Auto combat round 3
To me that is at the 3rd round, but could be seen at the second round. I have never seen on the other hand...
Bat
Auto combat round 1
Bat
Bat
Auto combat round 2.