Although many beginning band books don't teach this fingering in the beginning, it is a simpler fingering and should have little to no effect on intonation or tone. Considering the inherent complexity of beginning flute fingerings and the frequency of flat keys in early band literature, learning thumb Bb first is a valid decision.
This is usually the first fingering that beginning band books will teach. However, many teachers agree that it is an unnecessary added complication for beginning flutists. It is awkward compared to thumb Bb. Use this fingering when playing in keys with no Bb if that is more efficient than using the lever Bb shown on the right. It can also be used with a chromatic scale.
The lever key can be found to the left of your left pointer finger and is played by shifting your knuckle over slightly. It can be used when both Bb and B natural are required, similar to the one and one Bb.
Both middle and high C#'s are very sharp note on the flute. Without making any adjustments to the embouchure or fingering, the flute will play very out of tune. It is suggested that flutists try one of (or multiple) of the following:
blow down into the tone hole more than usual
roll in slightly to direct more air into the tone hole
add one or more fingers in the right hand until the desired intonation is reached (shown on the right)
Individual players and flutes will require different workarounds for this note. Experiment with different fingerings, embouchures, and air streams until you find the right match for you and your instrument.
Not every standard fingering is always the best choice. Depending on what the music requires, alternate fingerings, trill fingerings, or tremolo fingerings may need to be used. The link below will serve as a reference. On the right we see trill markings (above) and tremolo markings (below).