Yup we did originate from Rugby granted many rule changes. I like to make mention of Canadian Football that also originated from Rugby. It in many ways very simular to american football, both being Gridiron football. Its should be interesting to note that 1874 proto canadian football which was evolving form of rugby came to USA when game was played at Harvard. We adopted it and adapted it. Today American and Canadian football are simular but also very different. Both Rugby and Soccer started about same period 1820s due to differences in allowing hand touching.
It should be noted that orgins of football though many variations allowed hand passing. There then was split, one side who didnt want to use hands at all, thus becoming as we know soccer(english nickname), while others didnt mind using hands became known as rugby. I believe this is named after a school such was played at. Back in old days in USA Harvard was major player, with no offical leagues or anything when teams played, Harvard being harvard prevailed..
In the end calling american football in way is still correct. Football in correct sense has nothing to refer to how ball is contacted but from fact that ball was played on foot which was common for peasants vs on horses "Polo" played but upper class.
Okay history lesson aside, I still get a kick out of that image. By the way shape of the ball had to do with fact original balls were made from pigs bladders