Actually I believe it’s called “Football” because most ball games were originally played on horseback.

hmmm are you sure about that? I do not know with certainty but today playing a sport with a horse would be a high demand, back in the days it would have been an even bigger one so I sincerely doubt "most ball games" were originally played on horseback when not having a horse was that much easier

That's like saying "we call them Earthmobiles because back then, most vehicles were just planes" (no way planes were common before cars)

You are sort of right regarding the origin of soccer, but it was intended to distinguish football played by the association rules vs other types of football which were popular at the time.

From Wikipedia:

Association football is part of a family of football codes that emerged from various ball games played worldwide since antiquity. The word "association" in this term refers to the Football Association (the FA), founded in London in 1863, which published the first set of rules for the sport that same year.[8] The term was coined to distinguish the type of football played in accordance with the FA rules from other types that were gaining popularity at the time, particularly rugby football.[9] Heading from The Sportsman front page of 25 November 1910, illustrating the continued use of the word "football" to encompass both rugby and association football.

The term soccer comes from Oxford "-er" slang, which was prevalent at the University of Oxford in England from about 1875, and is thought to have been borrowed from the slang of Rugby School. Initially spelt assoccer (a shortening of "association"), it was later reduced to the modern spelling.[10][11] Early alternative spellings included socca and socker.[9] This form of slang also gave rise to rugger for rugby football, fiver and tenner for five pound and ten pound notes, and the now-archaic footer that was also a name for association football.[12]

Within the English-speaking world, association football is now usually called simply "football" in Great Britain and most of Ulster in the north of Ireland,[13] whereas people usually call it "soccer" in regions and countries where other codes of football are prevalent, such as Australia,[14] Canada, South Africa, most of Ireland (excluding Ulster),[15] and the United States. A notable exception is New Zealand, where in the first two decades of the 21st century, under the influence of international television, "football" has been gaining prevalence, despite the dominance of other codes of football, namely rugby union and rugby league.[16]