Kane was in an offside position, he received the ball from an opponent who deliberately played the ball. He therefore is not considered to have gained an advantage.
But yes, the laws with their application and interpretation are a mess.
This is not in the Laws of the Game.
Anyway:
Lovren did not deliberately play the ball. He would have to make a short backpass to play the ball. Miskicking or failing to kick the ball cleanly surely cannot be “playing the ball” in this context.
Imagine this scenario. I’m 5 yards offside, and the ball is played through to me. I am running on to the ball but I have not yet touched it. A defender comes alongside and tries to make a last ditch tackle before I touch the ball, he stumbles and doesn’t make a clean contact, getting only a slight touch. The ball continues to run into my path and I slot the ball coolly into the goal. It’s surely offside. If it was flagged offside then nobody would question it.
The sentence you mention is included to explain that if a defending player plays the ball to you and you’re offside, no offence is committed. Like Liverpool’s first goal, for example. That wasn’t offside.
Kane was offside and the offence complete *before* Lovren touched the ball. He was offside and committing an offence the moment his teammate released the ball. He was in an offside position and interfering with play. His presence meant that Lovren could not allow the ball to run through to the goalkeeper. His position, actions, and run were 100% interfering with play.