A flight instructor making a 360 turn on downwind at a non-towered airport reported a near miss with an aircraft on crosswind.

Narrative

Situation. While conducting pattern work at ZZZ; we had just entered the crosswind leg for Runway XX when we heard a transmission on frequency from an aircraft two miles south; inbound on the RNAV Runway XY approach with intentions to circle to land Runway XX. To provide spacing; I made a radio call advising that we would perform a right 360 turn in the crosswind to allow the inbound traffic to join the downwind without conflict; since we otherwise would have converged at the same point in the pattern. As we initiated the right 360; another aircraft in the upwind announced they would extend their upwind leg for spacing. However; as we were completing our 360 and preparing to re-enter the downwind; I observed that the upwind traffic; despite their earlier call; had already turned crosswind. This placed their aircraft on a nearly head-on course with ours. Upon visual contact and traffic alert on our Garmin; that aircraft immediately maneuvered left into the downwind; while I offset to the left and rejoined the pattern behind the traffic for safety. Causes. The upwind aircraft likely misidentified the circling RNAV inbound traffic as my aircraft while I was still executing a right 360 for spacing. Believing I was clear of the conflict or out of the way; the aircraft turned crosswind despite their earlier intention to extend upwind; creating a near head-on situation with our aircraft. Mixed operations between pattern traffic and inbound RNAV approach traffic created a congested traffic environment.Inconsistency between the upwind aircraft's stated intentions extend upwind and actual actions turn crosswindComplexity added by my right 360 maneuver; which may have made tracking my exact position more difficult for other aircraftSolutions. This conflict could have been prevented through clearer communication and stricter adherence to announced intentions. Had the upwind aircraft followed through on their stated plan to extend their upwind; proper spacing would have been maintained and the conflict would have been avoided.My decision to execute a right 360 turn after stating my intentions on the CTAF for spacing was an attempt to avoid conflict and build traffic separation for the aircraft on the RNAV XY circle to land XX. In future scenarios; extended crosswind or temporarily exiting the pattern may provide a clearer and more predictable method of maintaining separation when multiple aircraft are converging. As executing a right 360 maneuver may have made tracking my exact position more difficult for other aircraft.