What the weather was like before and after Trumps inaugural speech is irrelevant, and not mentioned in the article.

See the (speech) rain for yourself, plain as day, Trump's speech started about 30 minutes in, this video direct from the White House:

https://www.whitehouse.gov/inaugural-address

Trump's version -- "we’re not going to let it rain on your speech . . . it became really sunny. And then I walked off" -- is plainly wrong.  You have been given the irrefutable evidence that it's wrong.

Your video shows a lighter spot behind a cloud -- as Chief Justice John Roberts is introduced -- not Trump.