A Lacoste bag: Real or fake?

Hello, I recently acquired a nice Lacoste bag and I would like to know if it is genuine or fake :/

Here is my description: black bag, Lacoste logo at the bottom right (some threads are coming loose, is that normal?) sewn in a way that makes it look "placed". On the front side, an exterior pocket attached to the "body" of the bag by a button (you push it) on which "Lacoste" is engraved twice. On the back side, a simple pocket not attached to the body of the bag. At the level of the shoulder strap, it is black with white-gray edging. The bases of the shoulder straps (on the sides of the bag) are attached to the body by a "ring" identical to that on the front face, without the button (Lacoste also engraved twice). What connects the bases of the shoulder straps to the shoulder strap itself is a rectangular ring with rounded edges on which there is an engraving of a crocodile silhouette (no details, just the silhouette as the space is tiny). On the zipper, the crocodile is engraved respecting all the details of the genuine Lacoste logo (you can see all the scales, the tail is very slightly curved upwards, etc., perfectly conforming to the Lacoste logo that can be found on Google Images). The interior is smooth beige synthetic. The label troubles me though: above it says LACOSTE below a crocodile more or less (mostly more) compliant with the one found on Google Images. But below, a simple label, on which it says:
Exterior: Nylon
No interior
Made in China
On one side and the other the same inscriptions but in French.. I hope I have been as precise as possible. I can take photos if needed.. Thank you very much for a prompt response, it would help me :/