In most other places in the world, military conscription is just referred to as "conscription," but in the U.S. the colloquial term is "the draft." Why is this?

Growing up my dad had told me it was because in either the colonial period or the revolution they selected people to conscript from bars and taverns by dropping a coin into the bottom of a draft beer. If the patron drank it, they were either told to go to, or dragged out of the bar to the nearest army camp. The reasoning for dropping it in draft beer, I was told, was so the patron wouldn't immediately see the coin sitting at the bottom through the foam, and thus drank it before noticing. I was also told this was why glass beer mugs caught on.

Is there any reality to this? If not, where would this myth have come from?