So basically from everything I know and looked into when I researched this bag before getting a rep you find the placement difference between the different sizes.

The Kate in Small will have the logo at the bottom of the flap like your PSP shows or at least it did at some point. YSL may have moved the placement of the logos fairly recently as you will find the listing on the website currently shows it at the top where the Medium currently has it placed.

The Kate in Medium will have the logo at the top of the flap underneath the snap closure. I personally have not seen any authentic Kates will the logo at the bottom of the flap. I don’t think it had the placement change the Small version has had.