Flight Instructor reported an NMAC with a military aircraft on approach to another airport. The military aircraft performed a go-around to avoid collision.

Narrative

I was conducting a training flight with a student out of Camarillo Airport (CMA). After landing on Runway 26; we exited the runway and cleaned up the plane. While contacting Ground; my student became nervous during the exchange; and I took over the communication. Ground issued a lengthy instruction that included a request for our course of flight. Without verifying; I stated 'southwest departure.'For departure; I flew the upwind; crosswind; and turned downwind before proceeding eastbound toward our home base at ZZZ. CMA Tower instructed me to switch to Point Mugu Tower; which seemed unusual; but I complied. While monitoring traffic on my ADS-B display; I noticed rapid closure with an aircraft ahead and then visually acquired the traffic -- a small GA aircraft. I did not receive any response from Point Mugu tower; so I switched back to CMA Tower. At that point; CMA Tower advised that I was in conflict with oncoming traffic due to my eastbound departure rather than the southwest departure I had previously reported. And that I was also on course with a military aircraft that was doing an approach into NTD. The military aircraft was forced to go around and pass above me. I immediately acknowledged the situation; apologized to CMA Tower; and was instructed to switch to Point Mugu Approach.I believe this event occurred due to task saturation (managing a student on cross-country training); failure to double-check the assigned course of departure; and personal fatigue. Contributing factors included the busy airspace environment and my decision to rely on what I 'thought' the departure direction was; rather than confirming.This event reinforced the importance of verifying instructions; managing workload by prioritizing one task at a time; and maintaining clear situational awareness; especially when operating in complex or military-associated airspace.