What determines the attributes contained in a character style?

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When you change any value in the Style dialog, it is explicitly set in the style. You may see that happening by looking at Organizer tab, where those items will appear.

To stop some item from being defined in a style, you need to use Standard button, which resets all elements on a page - unfortunately there's no way to clear only a single element, so you'd need to set back those elements that should stay defined on the page after reset.

Also note that you might need to reopen the style dialog to see updated status on Organizer tab.