GA pilot reported a NMAC entering the traffic pattern when an aircraft flew straight in and directly overhead in while incorrectly stating they were entering on a 45-degree to downwind.

Narrative

On Day at about XA:05 pm above ZZZ airport; I was alerted from an aircraft holding short on the ground that there were two of us 'right above each other'. It was a near-miss.Prior to arriving at ZZZ Airport; I was transitioning northbound thru ZZZ1 Class Delta airspace which ends just 2 miles southeast of ZZZ Airport. From reviewing the incidents; I have gathered more information. The other aircraft reported that he was 'on a 6 mile final 25 left ZZZ traffic'. Since I was still on ZZZ1 frequency; I did not hear that call. I changed frequency from ZZZ1 Tower to ZZZ AWOS; then to ZZZ CTAF. I was aware of other aircraft in the pattern and on the ground at ZZZ Airport but; I was not aware of the other aircraft approaching from the east. I first became aware of the other aircraft when he reported 'on the 45 for the Upwind 25 left ZZZ traffic'. I looked ahead then to the north and northeast; but did not see the other aircraft. I reported 'yellow biplane about one to the east for the overhead approach; will be making a left break for landing'. The other aircraft reported 'Aircraft Y is upwind 25 left ZZZ traffic' which caused me to be very concerned since I did not see an aircraft ahead of me on the upwind. A Cessna that was holding short reported 'hey biplane; there's 2 of you guys right above each other'. I searched the sky all around me. As I looked forward again; I saw the other aircraft slightly ahead and barely above me. The other aircraft was moving forward away from me. I turned left and reported 'the Stearman is in the left break; not sure if the other aircraft is just passing thru or not'. The other aircraft reported 'Aircraft Y is turning crosswind'. I landed and returned to the hangar. Several days later I learned who the other pilot was.From Flight Aware; I learned that the other aircraft was not on a 45 to the upwind; but was behind me arriving on a straight-ahead descending course from the east directly to the upwind leg of the traffic pattern. The other aircraft overtook me from behind and flew very close over the top of my airplane. The other aircraft wrongly stated that he was on a 45.Before the near-miss; there was no direct communication between me and the other aircraft. If I had reported earlier that I was northbound out of ZZZ1 or if ZZZ1 Tower had alerted me of westbound traffic to the northeast or if the other aircraft would have reported when he was 2 miles east crossing the ZZZ1 freeway; we may have been able to de-conflict and avoid this nearly fatal situation. A contributing factor may be that the other aircraft was on a shallow descent and not able to see directly below the nose.