The confusion of terminology

Cubase LE AI Elements 9 and Cubase Elements 9 is it the same?

These are thre different Versions Cubase LE / Cubase AI / Cubase Elements. However they all shre the same installer file and manual. The license you bought, determines which version will run, and therefore which features are available.

I purchased the full version of Cubase Elements 9. Shortcut to run it Cubase LE AI Elements 9. Help - about Cubase LE AI Elements. This is normal?

As long as the license in elicenser control centre shows elements then it is correct.