Say I have a definition and I am trying to find an English word that represents it, and I provide an example of said word (think word=fruit, example=apple). Someone gives me a word whose definition is a superset of mine (think food). How would I go about expressing "My example is contained within the definition of the word you suggested."? contained does not seem very sound here. I am looking for a word that is used commonly.
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"Contained by" is a reasonable enough choice.
A more precise usage would be
These each have precisely the meaning you are seeking, that a word defines something that is contained within the definition of another.
Neither are particularly common words though. If this distinction comes up often for you, they are worth learning, but if it's important, as you say, to use a common word, then I'd say keep with contained; it's not as precise but that's the cost of keeping to more common words.
You could say your example is included in his definition, or that your word is a subset of his word.