After you select skip tutorial it teleports you to the center of the town. Does this count as a warp?

yes

yea it counts as warping if you teleport there
runners tend to use TSS to skip the tutorial with the button but without teleporting

WTS (Warpless tutorial skip) can only be performed in versions RC1-RC5. WTS is the only way to skip the tutorial without a warp. In one of these versions, you have to go up to the tutorial red gate and hold space as you roll through it. This will entirely skip the tutorial sequence and let you through without warping. Skipping the tutorial in any other way is a warp.

How do you get RC1-RC5?

https://www.speedrun.com/island_games/resources