I see no indications of the player letting you change lane in front of them. I see a player keeping a safe distance to the player in front. At this place, the player has right of way since it is considered a lane change and not a zipper merge. This is indicated by the lines following through the entire length of the lane and not ending before the lane ends.
This also means you had to give way to the player.
Due to lag the player might have been even further forward on his own screen and you might have been close to colliding with the player.