Flight Instructor with student reported a NMAC at AUO non-towered airport when another aircraft entered the pattern and flew directly toward them. Flight Instructor took evasive action to avoid a collision.

Narrative

My student and I were on a practice IFR flight doing approaches; specifically the RNAV 29 Circle 18. We were 5 NM out on the final approach course; getting ready to circle to Runway 18 when I see a plane off to my right but he was reporting a midfield left downwind when he was actually more of a left base. I communicated with him and asked for his intentions. He kept saying midfield when he just was not. He started turning in towards us to come in behind us but he was just coming in towards us at the same altitude. I had to talk to him again and he still couldn't see us; so I had to take controls from my student because I was the Safety Pilot. We broke off the approach and had to reintercept it back. We had the traffic alert pop up on our MFD (Multi-Function Display); and I had to take evasive action. After he spotted us; he followed us in way too close for my comfort. It felt like tailgating essentially. He was less than 0.5 NM from us the whole time in the pattern all the way to landing.I want to suggest better rules for pattern entry as well as pilots knowing better position reporting capabilities. Auburn is a busy training airport and non-Auburn general aviation pilots tend to not realize how chaotic it can be at times. A lot of traffic conflicts occur with Auburn airplanes and non-Auburn airplanes; and it needs to be addressed.