A corporate jet pilot departing a non towered airport reported a NMAC with an aircraft entering the traffic pattern.

Narrative

I announced my taxi from NE hangars A2 A RWY 19. An aircraft taxi out announced they would hold for us even though they were not a factor as spacing was good. Holding short for departure for runway 19 we held for a landing aircraft as well as another aircraft. We listened to aircraft communicating on 123.075; and Aircraft Y reported they were approaching from the SW; would cross over mid field; circle for the 45 and left traffic for RWY 19. I could see them approaching from the SW on TCAS. After the landing traffic (not the aircraft involved in near miss) announced they were clear of the runway exiting taxiway A2; I announced we were lining up on RWY 19. On the RWY I announced we were departing RWY 19 and would make a turn to the left to avoid the traffic from the SW. after rotation I noticed this aircraft was not tracking for mid field and instead the departure end of the field. I started my left turn early to avoid this aircraft. I also shallowed my climb in an attempt to avoid the converging traffic. Despite my announcements the instructor of the other aircraft said they did not hear me - although other aircraft had heard and responded (an aircraft holding on taxiway Alpha). When I asked if they saw us; they said they did but they made three communications and didn't hear ours. I am not sure it makes sense to ever fly over the departure end of a non-towered airport and it was different from what said in their communications.