General aviation pilot reported a NMAC during initial climb with opposite direction inbound traffic.

Narrative

I was on a training flight in the traffic pattern at WVI using runway 20. Aircraft Y had announced they were on the RNAV approach runway 02 into WVI terminating in a missed approach procedure. We were touch and go on runway 20 on the departure leg. I established visual contact with the inbound Aircraft Y and announced to their tail sign with our tail sign as well that we would side step and off set toward the west of the runway and allow them to fly the published missed approach over the missed approach point which was the runway threshold. They acknowledged our call to them. I began a turn toward the west and offset of the extended centerline. As I was turning; the same Aircraft Y began to turn toward the west and continued their descent toward us rather than continue toward the missed approach point. I maintained my west bound turn and leveled off due to terrain avoiding Aircraft Y. I called them up again on CTAF and sounded to not be aware of the situation. Upon landing; I checked ADS-B data and at our closest point; they were 100 feet above and 350 feet off set of us toward the east far off the final approach course for the approach.