What matters for most purposes of daily life is the solar day, which is 24 hours by definition. This is the time it takes for the Earth to fully rotate with respect to the sun. A sidereal day is how long it takes the Earth to rotate with respect to more distant stars. This is shorter because it also includes the motion of Earth around the sun (which is why you found the accumulated yearly difference to be exactly one day!)
Leap years are to correct for the fact that a solar *year* is not exactly 365 solar days, but rather close to 365.25. Therefore, every 4 years,* we’re one day further around the sun than we were, so we add a day to compensate.
*It’s not *exactly* every 4 years because a solar year isn’t *exactly* 365.25 days. On years divisible by 100 (except those also divisible by 400), we skip a leap year.
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