GA pilot reported an NMAC while on the downwind at non-towered EDU requiring evasive action. The pilot of the other aircraft had reportedly not been communicating properly; which contributed to the event.

Narrative

While on upwind in the pattern for Runway 17 at EDU; a faster airplane reported 4 miles inbound from the south. I assumed he would make a standard 45 entry and join behind us in the pattern; so I continued onto the crosswind; reported my position; and then turned onto the downwind. I was surprised when the other pilot reported that he too was on downwind; so I reported my position (on downwind abeam the 17 numbers) and began my descent. The other pilot did not clarify his position; and at that moment I saw an advisory on my ADS-B device that he was directly below me by 200 feet; evidently having already started his descent. I immediately arrested my descent; added power; and climbed back to TPA while bailing out of the pattern to the east of the field. I felt this was the safest direction to fly; in order to fly clear of the pattern; rather than to risk getting even closer to the other aircraft should he also choose to take evasive action. After reaching pattern altitude; I maneuvered over the university campus east of the field; at a TPA of 900 feet MSL; to re-enter the pattern on a left 45-degree entry while staying within 2 miles of the airport. It was then I realized that I was 100 feet below the minimum altitude required for VFR flight over a congested area. Even though I was within 2 miles of the airport; I now realize that I should have climbed to 1;000 feet AGL until entering the 45-degree leg to the downwind. I think this occurred due to getting saturated with what appeared to me like a very hazardous situation with a faster-moving aircraft who didn't make adequate radio calls; suddenly catching up with me in the pattern. In the future; I will anticipate that pilots may not make accurate calls with respect to their range and to seek clarification in cases like these.