Games in which you are eliminated do not count toward the non-premium game limit regardless of whether they appear in the "My Active Games" list.
The "ignore" button is for removing them sight (to help expunge the traumatic memory, I suppose), but it does not affect the number of games you can play in, as the games in which you are eliminated were already ignored for game-limit purposes.
"As a general truth, communities prosper and flourish, or droop and decline, in just the degree that they practice or neglect to practice the primary duties of justice and humanity." WHS
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