GA pilot reported an NMAC in the traffic pattern when another pilot mis-stated they were on base leg but turned crosswind in front of them.

Narrative

NEAR MISS at ZZZ on Day at XA:07 PM between Aircraft Y and Aircraft XAircraft X was on an IFR flight plan from ZZZ1 to ZZZ on Day. Weather at ZZZ was MVFR (2;200 ft; overcast with winds at 340 at 4-6 kts). Based on weather and winds at ZZZ2; Aircraft X advised ATC that would like the RNAV XX approach. ATC vectored Aircraft X to ZZZZZ intersection (IAF for RNAV XX) at 3;000 feet then cleared Aircraft X for the RNAV approach. Aircraft X was monitoring Unicom and noted that Aircraft Y had just departed Runway XY from ZZZ. After ZZZZZ; Aircraft X proceeded with the RNAV XX approach and descended to 2;600 feet to KEENO and then continued the approach to Runway XX at ZZZ. Shortly thereafter; Aircraft Y announced that it was on final for Runway XY at ZZZ for a touch and go. Aircraft X became VFR and cancelled the IFR clearance with ATC and proceeded VFR. Aircraft X then announced to Aircraft Y that Aircraft X would change and enter a downwind for Runway XY. At this point; Aircraft X did a slight jog to the right to line up with the left downwind leg for Runway XY and to reduce the potential for a head on collision. Aircraft X continued on the visual approach. About 2-3 minutes later; Aircraft X announced that it was on left downwind for Runway XY at ZZZ (at 1;700 feet MSL). Aircraft Y announced that it was on base leg for Runway XY at ZZZ. Then the near miss occurred. Aircraft Y appeared directly in front of Aircraft X; less than 500 feet away; in a climbing; left turn from Crosswind to Downwind. (Distance was estimated by hold hand out at arms length; palm facing pilot; and noting that aircraft was still visible on left; right; and top of hand. Aircraft X immediately slowed up and mentioned to Aircraft Y that he was on crosswind; not base and stated and caused a near miss. Aircraft Y then responded; 'oops - yeah - I meant to say base - sorry meant to say crosswind'. Aircraft X advised Aircraft Y that it was behind him and would follow in trail. Aircraft Y continued in the pattern and did another touch and go. Aircraft X landed without incident.After 10-12 minutes; Aircraft X was ready to depart ZZZ again and noted that the winds were still at 340 at 6 kts. Aircraft Y had changed direction and was now in the pattern for Runway XX at ZZZ. Aircraft X waited for Aircraft Y to land and then (while both aircraft were on the ground); Aircraft X advised Aircraft Y that it was going to file a safety form for the near miss. Aircraft Y simply stated that 'you were behind me - do whatever you need to do'. It appeared Aircraft Y was never aware that he cut directly in front of Aircraft X and due to his level of bank; would not have been able to see Aircraft X. Aircraft Y did not apply any principles of situational awareness and nearly caused a mid air collision. It was also very disturbing that the pilot of Aircraft Y did not want to discuss the situation at all.