I've tried Elmer's white glue, contract cement gel and am trying a water based contact cement soon.
They have their nuances .
It's easier to work with white glue thinned with water.
Removal can be a pain b/c the leather really sticks to the wood. Needs heating to remove the leather cleanly. If the handle is not cored, it might warp if heated.
Contact cement has a smaller margin for error. Some thin it out and do overnight setup/drying.
Removal and clean-up is easy on contact cement gel. But, it's another toxic chemical in the shop. Smells nasty and has nasty fumes.
I did not know glue stick could be used for leather wrap until GB mentioned it.
For leather, I use weldwood contact. Works well, IMO. Have never tried to do Lizard, or use elmers for leather, only linen, and now use the elmers glue stick for linen
Dave