Both renamings are in the primary record. Macrobius, Saturnalia 1.12, preserves the decrees: Quintilis became Iulius in 44 BC because Caesar was born in it; Sextilis became Augustus by senatorial decree in 8 BC. Suetonius, Augustus 31, gives the stated reasons: in that month he first entered the consulship and won his greatest victories, Egypt among them. The month you call August is, on paper, a war memorial.
Everyone knows the story: July had 31 days, so Augustus took one from February to keep his month from being shorter, and that is why February limps. It is a perfect little tyrant anecdote, and it is not in any Roman source. Macrobius' account of the calendar has the month lengths fixed before the renaming; the story of the theft surfaces in medieval computus handbooks, a thousand years late.
Of all months, the Senate chose Sextilis because it held the climax: Alexandria fell on the first of it in 30 BC, Antony and Cleopatra died, and the last Hellenistic kingdom became a Roman province. August is not named for a man's birthday; it is named for the month the republic's long war actually ended. Every August you write on a form is a dated anniversary of the annexation of Egypt.
Every emperor after them wanted a month and nearly all failed. What does it say that the only two names still on your calendar belong to the men who never had to rename it themselves?