'Horse of another color' is the opposite expression from Shakespeare's 'a horse of that color' in the Twelfth Night Act 2 Scene 3 where Maria told Sir Toby Belch that her purpose was indeed a horse of that color which meant that Sir Toby Belch got the same idea as hers. So, 'horse of another color' is an idea different from the other.
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Twain accomplishes his purpose chiefly through telling a story within a story, also called a "frame story".