Corporate flight crew reported runway incursion during takeoff climb at a non-towered airport.

Narrative

Communicated with Phoenix Departure 121.9. Given a Takeoff void time of 5 minutes. Using Unicom frequency 122.95 transmitted to traffic that we were departing Runway 21 for a South bound departure. Taxiing onto Runway 21. I (Pilot Not Flying) visually cleared final and departure end of the runway. After rotation at about 400 ft. Pilot Flying noticed landing lights of traffic on opposite end of the runway slowing a roll. We increased climb and vertically cleared opposite traffic by 800-1000ft. Upon evaluation after incident I realized we were transmitting our airport position through UNICOM frequency rather than CTAF. No radio contact was made with other aircraft during or after incident Runway 21 Approach lights on the displaced threshold made any aircraft lights at the opposite end almost indistinguishable. Unable to verify if opposite traffic had entered the runway before or after our roll commenced.What could've prevented this situation is use of correct CTAF Frequency to properly communicate with traffic in the area. Proper verification of frequencies for non-towered and night operations going forward will help minimize this situation. Better scan of runway clearance; especially long runways; to ensure runway is clear for takeoff or landing.