C182 pilot reported entering the airport traffic pattern at the wrong altitude and direction resulting in a NMAC with another aircraft executing a go around at the airport.

Narrative

This report involves an incident at ZZZ where I was the pilot in command. The incident involved a near miss (from the perspective of the CFI in the other aircraft that was involved). The incident occurred on Day 0 at approximately XA:30. Weather was VFR; light winds and clear skies. Barometric pressure was 30.30. The flight originated at ZZZ1 and was for the purpose of ferrying an aircraft from ZZZ1 to ZZZ. I used flight following for the flight. I flew at an altitude of 5;500 and stayed east of the freeway to avoid entering the ZZZ2 Class Charlie airspace. I was vectored a few times by ZZZ [TRACON] to avoid traffic going into ZZZ2. Once past the traffic I was given the clearance to navigate toward my destination. Flying into ZZZ requires the pilot to announce their position and intentions. I was approaching ZZZ from the northeast and my plan was to cross midfield and enter the pattern for left traffic for Runway XX. I announced my position at 10 miles out; 6 miles out; 4 miles out; 3 miles out and then every 10-15 seconds as I approached the airport. There were several other aircraft in the pattern and I was aware of an aircraft that was coming in for landing on Runway XX. That aircraft was not announcing their position or intentions. That said; I had eyes on the aircraft and maintained visual contact with the exception of a few quick glances at the airspeed indicator and altimeter. As I was getting ready to cross midfield I announced my position and intentions once again and called out my altitude at 1200 over midfield. However; I was quite fixated on the aircraft on the short final. It is very possible that I drifted toward the departure end of Runway XX by being 'fixated' on the aircraft on short final. I am fairly certain that my altimeter read 1;200 feet though looking at flight aware I'm indicated at 850 feet close to the departure end of the runway. At this point I noticed the aircraft on short final execute a go around. I maintained visual contact with that aircraft and did not feel the need to execute an evasive maneuver given the separation between us. I judged from above that there was enough space between us. I proceeded to enter left traffic on the downwind and landed uneventfully on Runway XX. The other aircraft did not communicate intentions or say anything on the radio. I am filing this report; however because I have learned that my approach into the airport was not the recommended procedure and also because I realize that my altitude and approach were likely not within regulations. I have some ideas on how to mitigate this type of event in the future and several lessons learned. First; the majority of my experience is at towered airports. I would say that at least 98% of my landings and air work have been at towered airports. I am quite used to entering mid field to enter the pattern via ATC instructions. I have applied this same approach to untowered airports when approaching from a direction opposite the pattern (in this case from the northeast). I discussed the incident with my CFI and was informed that my approach was incorrect. The CFI pointed out AC-90-66 on proper procedures for untowered airports. I was surprised to learn that my approach was not appropriate. I recognize that it is my responsibility to adhere to all AC and that I should have known the recommended approach. Second; I also realize that my fixation on the other aircraft likely caused me to drift and not cross midfield as intended. That also likely contributed to my being lower than the pattern altitude. This reminds me of the old adage: aviate; navigate; communicate. So there were two errors on my part: wrong approach and wrong altitude. I realize that this could have caused an unsafe environment for the other aircraft even though I felt there was enough separation. From the other aircraft's point of view they were lower and performing a go around and it likely looked worse than it was from their vantage point. Had I beenhigher I think it would have created a bigger margin of safety and I understand why it's important to calibrate altimeters; and to maintain situational awareness. For my part; I have read the AC and plan to take a refresher course via the Wings safety program on untowered airports and plan to go up with a CFI to ensure I can practice the procedures for these type of airports.