General aviation pilot reported they descended below pattern altitude too far from the airport after misunderstanding the Tower Controller's pattern entry instructions. Controller vectored pilot back and pilot completed a touch and go landing.

Narrative

Please note that this is per my best recollection. My intention is not to cause any trouble for any party; as the solution was resolved positively. My goal is to simply inform of how I as a pilot will respond to this in the future. The goal of this flight was a time building flight for cross country hours for my instrument rating via Part 61. I had VFR flight following from ZZZ1 and was handed off from ZZZ Approach to ZZZ Tower. I was a little nervous about this flight as it was the first time I had flown solo to the field. I informed Tower that I was 10 miles inbound touch and go; with a departure to south to follow. I was instructed to contact at 3 miles out and plan for a left downwind for Runway XX. This approach different from the other flight I'd done to ZZZ; which was on the larger Runway XY/XZ. I was approaching from near due south (northbound heading toward ZZZ). As I neared the 3 mile distance from the field; immediately prior to informing Tower that I was at 3 miles; per requested; Tower preemptively informed me to enter a left downwind Runway XX. Rather than enquire if they are instructing me to do so immediately; or simply advising of how they want me to fly the pattern; I read back my instructions and entered a left downwind for Runway XX. This resulted in a wider traffic pattern than I am typically used to flying - and one that is a bigger than what the FAA airplane flying handbook recommends. I had the field in sight; though terrain features and the condition of Runway XX made it much more difficult to consistently make out relative to the larger Runway XZ at a 3 mile distance. I was advised of other traffic in the approach phase for Runway XX; which split my visual duties between maintaining visual contact with Runway XX and visual contact with other traffic. Abeam XX I began my descent below traffic pattern altitude and turned base following landing traffic on a 4 mile final. Tower then radioed me and told me to turn base; which caused confusion as I felt I was already on a base leg (cross checking with my HSI (Horizontal Situation Indicator) and GPS extended runway centerline confirmed this). Tower informed me that I had descended below traffic pattern altitude of 1600 feet - which cross checked with my altimeter - however I realized I must have been too low for the distance of my (extended) base leg from the runway (altitude was approximately 1200 feet MSL). At this point the instructions I received from Tower did not fully align with my expectations of a stabilized approach. While I felt that I had Runway XX in sight - I informed Tower that I wished to abandon the approach and try again; that I was new to the area; and that I had lost consistent visual contact with Runway XX (I still had contact with the runway; but the information I was receiving from Tower made me doubt that I had full situational awareness of where my plane was approaching...if it was the correct runway or if I had some sort of expectation bias going on. Something just didn't add up). Shortly following this request; Tower inquired if I could see the approach end of the Runway XX - which I said yes - I had just then watched another plane touch down on Runway XX confirming that my approach was indeed pointing at the correct runway. He asked if I made a right 270 across XX's final if I would be able to descend and land - which I said yes. I was cleared for a right 270; lined up with XX's final; and performed a touch and go. I thanked Tower for helping me regain situational awareness and departed the area with an uneventful trip back to my home field.This root cause of this situation occurred when I entered a downwind leg 3 miles from the field with further degradation to situational awareness and confidence in a stabilized approach occurring as the approach continued. This caused confusion on my end as I had to work harder to maintain a 100% positive visual identification with Runway XX while also looking for traffic at a distance further than usual. In the moment; it felt as though ATC had instructed me to enter a downwind leg at a 3 mile distance from the field; but the reality is that they were instructing me to enter a normal downwind leg (and that instruction came when I was at the 3 mile mark from the airport). In the future - I intend to ask for clarification from ATC as soon as I feel I receive an instruction that does not align with the norm (entering a larger traffic pattern; for example). ATC did an excellent job identifying that my approach had momentarily shown signs of slipping from a stabilized criteria.