Americans say "Elmers" because the brand is recognized throughout the US, just like "Xerox", which is now both the brand and the process. Any washable glue will work. I have used both the blue gel (stick) and the white stick and liquid and have had equal success.
I remember it also coming in a paste form. The teacher would spoon out about half a teaspoon onto a piece of construction paper and we used a popsicle stick to spread it. And there was always the kid who would eat his. Ewwwww!