Actually, it was originally called soccer by the British. In the 1860s, there were a number of sports called "football," and so they acquired different names/nicknames. So for example, rugby was generally referred to as Rugby Football. During that time, what is now modern soccer/football was the result of a number of teams getting together and unifying all their varying rules, which they then called "Association Football."
Now, the nickname of the time was to call rugby "Rugger." Because of this, "Association Football" acquired the nickname of "Assoccer." Which was rapidly replaced with "Soccer."
As to your class statement, it's not nearly that simple. Both rugby and soccer were originally upper class sports in their organized form. Soccer caught on with the lower economic classes more so than rugby, and it was at this time, nearly 20 years later, that the formal name "Association Football" went a different direction and became simply "football" to your blue collar Brits.
There is actually a British saying, “Soccer is a gentleman’s game played by ruffians and Rugby is a ruffian’s game played by gentlemen.” That said, your statement about it being called football because it was played on foot rather than mounted is strictly correct, it just doesn't apply to the particular evolution of the modern sport.