One of the FallaciousArgument''''''s, wherein multiple definitions of the same word are                                                                             use by the speaker, without clear separation between them.  Often times, a specific term with a strong negative connotation is applied to a larger group (who share some attribute with the group that the specific term refers to), with the intent of applying the negative connotation to the larger group (for which it otherwise would not apply).

Examples:  "<insert politician here> is a socialist because he supports <insert leftist policy here>, a policy supported by <Stalin|Mao|Pol Pot> who murdered ''x'' million people".  The implication is that because the politician supports a given policy--that makes him a "socialist" and thus a fellow-traveler of all of the various Marxist totalitarians the world has known.

Similar arguments are also made about politicians on the political right (and elsewhere on the spectrum, too).

The fallacy is obvious--having one point of agreement with someone doesn't at all imply agreement on all points.

See also LanguageAbuser
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Your example is not one of Equivocation but one of GuiltByAssociation, this is an example of equivocation

Bowling balls are not light (heavy) therefore, they are dark. In this example the word "light" is used in two different contexts.

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All skaters are people.  Tonya Harding is a good skater.  Therefore, Tonya Harding is a good person.

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I'm a grammar nazi.  My brother is a grammar vandal.  Nazi's [sic] are a German party created in 1919 and took over much of Europe by 1940.  Vandals are an East Germanic tribe that sacked Rome in 455 and dwindled out of existence by 534.  Therefore: My brother and I are German, my brother is at least 1400 years older than me, and we're probably planning to take over the world... or at least ransack it.

No, no... My brother corrected me.  I am a grammar gestapo.

''Shouldn't that be "Nazis" (or better still "The National Socialist Party"), not "Nazi's"?''

Indeed, it should be 'Nazis' or ''Nationalsozialisten''.  And that was a test.  You're welcome to join the party.  We must defeat the grammar Vandals!

''Surely the grammar Goths would take exception to this?''