At this time of the day, Mutt will be hard at work in his day job ... so I'll jump in ...
"take-off flare distance"
read "distance from lift off to screen height" .... engineer-speak (especially if one comes from an aerodynamics or FT background) ... please tolerate our occasional use of in house mumbo-jumbo jargon.
first segment (rotation and gear retraction)?
first segment is the distance from the screen height to the completion of gear retraction .. and may, or may not, exist .. depending on how fast the retraction sequence is.
Are you saying that the first segment has to be completed over the runway
not at all .. Mutt's earlier comment -
Requirement to have half of the takeoff flare distance over the runway
refers to the take off run required (TORR) part of the takeoff calculation which (for most jurisdictions) requires at least half of the distance to screen to be over the declared TORA. Simply put, this tries to avoid too many instances of dust, dirt, and grass blowing around the aircraft at the far end of the runway ...