It probably helps to think of it as "usage" not "definition." Both "spoiled" and "started" usages for "off" are slang. I find this perspective useful anyway, because if I can think of the original definition or usage of a word then it's easier to map it to a word in a different language. Many times there are words that came from the same root and knowing the foreign usage of the word of the same root helps me better understand the meaning of the English word. I'm a bit nerdy like that.
Actually, while spoiled is slang, the "started" definition is an official dictionary definition. And there is in fact over 50 official definitions of "Off" according to dictionary.com and a similarly large list from the oxford dictionary. So that may make the whole "remembering the original definition or usage" a bit trickier.