Short answers, pulled from the story.
A woodwind instrument is defined by the mechanism that splits the air to create vibration rather than the material of its body. The classification remains rooted in the method of sound production where air is forced against a sharp edge like a reed or fipple to generate tone.
Open flutes operate by directing a focused stream of air across a sharp edge to split the airstream against the air column inside the tube. Closed flutes function differently by requiring the player to blow air into a duct that channels the breath to a sharp edge with precision.
Reed instruments trace their origins back to the middle to late Neolithic period following the observation of wind blowing through a split rush. Sound is produced by focusing air into a mouthpiece that causes a reed to vibrate and set the air column inside the instrument into motion.
Saxophones are included in the woodwind section because they require a reed to produce sound rather than relying on the material of the instrument. The classification overrides the visual evidence of their construction materials to unite diverse instruments under a single historical banner.
The modern orchestra woodwind section typically includes flutes oboes clarinets and bassoons with supplementary instruments like the piccolo cor anglais and contrabassoon. The concert band woodwind section is significantly larger and incorporates alto tenor and baritone saxophones alongside standard orchestral instruments.