Tower Local Controller reported a departing aircraft flew over another aircraft taxiing across the runway.

Narrative

Working the Local Control position; Aircraft X advised ready for departure Runway 10L. My Ground Controller asked if I could accept an opposite direction departure Runway 28R at [Taxiway] B intersection. I said yes. I cleared Aircraft X for takeoff Runway 10L and visually scanned the runway. I observed Aircraft Y taxiing up to the [Taxiway] B intersection hold line on the north side of Runway 10L/28R; which is normal operation. I glanced down to the (Information Display System) IDS4 to verify the weight category of the Aircraft X reference wake turbulence for the opposite direction departure from an intersection. I heard 'Tower did you see that?' on frequency and replied 'say again.' As I did; I looked up from the IDS4 to see Aircraft X airborne; approximately 50 ft. off the ground and roughly 6000 ft. down the runway (just east of Taxiway D. Aircraft X replied 'looks like a turboprop just taxied across 10L.' I looked down to the east end of the runway to see Aircraft Y across at [Taxiway] B and turning east onto Taxiway A. No coordination had yet been done with Ground Control to approve a crossing. Clarify taxi/hold short instructions reference Taxiway B on the north side to the tanker base ramp. This area is very confusing as to the length of Taxiway B and where the movement area vs non-movement ramp area are. I don't believe there is enough of Taxiway B to warrant a taxi instruction on the north side; but rather just 'hold short of Runway 10L/28R at Taxiway B.' Once coordination has been accomplished for a cross; then 'taxi via B...' would be applicable.