PC-12 Captain reported an unfamiliar local pattern entry procedure assigned by ATC caused confusion and miscommunication leading to an airborne conflict with other aircraft.

Narrative

Prior to approach to ZZZ; cancelled IFR to proceed VFR following [Interstate X] through the canyon and [Interstate Y] northbound. We (the crew) have flown this approach multiple times before; typically directions are follow [Interstate Y] north and turn a left base to final for approach to Runway XY at ZZZ. ZZZ Tower asked; 'Are you familiar with the racetrack pattern?' Both of us; having read numerous articles on 'racetrack patterns' (rounded base to final) recently and having flown very similar approaches to the same runway before; agreed that we were familiar and responded 'affirmative.' I believed we were given a descent to pattern altitude; followed shortly by a clearance to descend from 9000 MSL to 6000 MSL and 'follow the racetrack pattern [Interstate Y] northbound; [local road].' Shortly after initiating descent with gear and 15 degrees of flaps; TCAS alerted to a traffic conflict. We saw a Cessna 172/182 (close enough to read the tail number and remember the paint scheme) pass 12-2 o'clock maybe 500 feet lateral separation very shortly thereafter which prompted me to make a left turn to avoid and increase separation; ATC provided no advisory of this traffic before or after TCAS alert (inside Bravo airspace). Descent continued into rounded left base angling toward final to Runway XY.In hindsight; no traffic sequence (a la 'you're number two; after you complete a left circuit you'll be following a 737') was given; so it was not immediately clear that the 'racetrack pattern' the tower desired was to essentially fly a displaced rectangular course east of the airport (or that traffic separation was required for other traffic landing ahead of us). (This was clarified with ATC on the phone in response to the subsequent warning; they named a road as part of this pattern which they have mentioned before when we've flown a traditional pattern and repeated '[Interstate Y] north; and I think that caused the unknown confusion in the air.) While angling toward a final for Runway XY; right seat pilot noticed one of the two airliners off the right was actually on final to XY; not XXL as initially believed. In response; we turned farther left (east/southeast) as we saw the 737 begin to initiate a go-around. We leveled off around 6000 headed southeast bound and asked the tower for clarification and further instructions. They gave us a left turn to loop around and rejoin [Interstate Y] north; then cleared us for a left base and landing on XY. After explanation via phone call; the tower's initial directions were clarified for future use. While I acknowledge we inadvertently failed to comply with their directions; I believe the following factors contributed to the traffic conflict with the 737: 1) lack of a traffic call-out for the Cessna at the top of descent; resulting in distraction and a preemptive left turn; 2) overly complicated directives from the tower with the use of an ambiguous term ('racetrack pattern'); 3) failure to maintain cleared altitude (6000) due to confusion over landing sequence and perhaps failure to clarify directions we initially thought we understood."