General aviation instructor reported a near miss with another aircraft at a non-towered airport; during a training flight on takeoff and during the climbout. The instructor identified another aircraft going around while starting the takeoff roll; then the instructor took control of the aircraft in the initial climb; and maneuvered away from the go-around aircraft.

Narrative

I made a bad judgment call and did not asses a situation as well as I did which led to pattern traffic having to execute a go around while we were taking off. What was our plan:-We were conducting a complex/high performance training flight in the retractable gear aircraft. Our briefed plan was to takeoff out of Harvey for an eastbound departure and maneuver east of lake Stevens for complex/high performance training before conducting full stop taxi backs at ZZZ for landing practice.What happened:-We were holding short of 15L and noticed traffic from our perspective abeam the numbers in the right hand downwind.-This seemed like plenty of time to depart in front of traffic as this seemed like a routine pattern.-Me and the student made our takeoff call while pulling out onto 15L. -As we were pulling out onto 15L; I noticed that the traffic started a short approach and this started to look more like a power off 180. -Traffic called base turn and shortly after final as we were lined up on 15L.-We applied takeoff power because at the time it seemed like the safer option to get things moving and clear the runway for the landing traffic rather than sit there being a hazard.-after takeoff a few things happened that added a lot of confusion to this scenario. traffic on final said something along the lines of 'were sidestepping to the left of the runway'. We never heard the standard phraseology words 'going around' which led to us having to physically look outside trying to confirm if traffic was going around.Our GPS screen with the ADS-B dimmed automatically to the lowest brightness value literally right after takeoff. We had limited visibility on the traffic because they were in our blind spot and we were in theirs... High wing vs Low wing. (I'll get back to this)Jefferson county radio calls were going on at the same time which made the cockpit noisy and hard to tell who was saying what. The issue we caused-We caused a very hazardous situation. We are taking off while an airplane is going around. Same flight path. Not good.What reactive actions did we take:-I assumed the airplane did go around and was sidestepping to the left so I took flight controls and at a safe altitude changed our plans instead of turning left into the conflicting traffic; to depart right hand to the north west away from traffic on the go-around. Instructed the student to make a radio call communicating that we were doing so and that we have go around traffic in sight.Reflection-We had a long debrief after our lesson about communication. This would not have happened if we anticipated a short approach and communicated based off of that knowledge. We would not have lined up on the runway. Up until we were on the runway I perceived a routine pattern until I started getting the picture behind us that they are closer than I was picturing/anticipating.-The radio was very busy during this situation at Jefferson county and people were stepping on each other. I heard that someone was sidestepping to the left and I concluded that must have been the traffic on final doing a go-around. Like previously stated the ADS-B display dimmed to zero after takeoff. I wanted us to be as predictable as we can so we just kept climbing straight out and kept calm. After I took flight controls; I saw the airplane off our left and we were on diverging flight paths.