Most sources (including Guinness World Records) agree that the highest clouds in the Earth's atmosphere are noctilucent clouds. Latin scholars will not be surprised to learn (if they didn't already know) that these are only seen after sunset. They form at altitudes of around 50 miles – "above 99% of the atmosphere" – and are seen when they're illuminated by the rays of the setting sun.
Noctilucent clouds are too high to affect the weather, which occurs in the lowest regions of the atmosphere. But even in those regions, it's not correct to single out Cirrus as the highest of all. As this diagram on Wikipedia shows, Cirrocumulus and Cirrostratus clouds are at the same sort of altitudes (around four to six miles), and Cumulonimbus can (if the diagram is to be taken literally) reach as high as seven miles.
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