A320
Flight Crew Air Transport Pilot (ATP)
Situational Awareness; Confusion
ATC Issue All Types; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance
Flight Crew Became Reoriented; Flight Crew Returned To Clearance
Human Factors; Procedure
Clearance was ENEKO MARUX V4 SJD.  While descending into MMSD; we were on V4 between ENEKO and MARUX; then after MARUX; given clearance to descend to 6000. Approximately 20 miles from the SJD VOR; we were cleared the VOR DME-2 Rwy 16 approach.  The FMS was programmed from MARUX to the D11.0 fix; instead of including the the IAF (which is the DME 15.0 fix).  The approach plate is confusing; and led us to believe we were cleared to descend to 3600 when we had to wait until the IAF; which was not programmed correctly into the FMS.  ATC informed us that we were descending early and below his MVA.  He cleared us to level off; and continue on a visual approach.  We were in VMC conditions the entire time.  The two aircraft arriving before us were given headings to intercept the approach; and the controller was very difficult to understand; and was unintelligible at times.  I was the pilot flying and this was also my first time flying into Cabo; as well as my only international flight since I completed training. Those factors combined with unclear approach charts (especially depicting the transition from the encounter structure to approach); as well as the controller stating 'cleared for VOR DME-2 Rwy 2 Approach' without saying 'VIA MARUX'; led us to omit the IAF during our briefing and approach.  I believe this could have been avoided by having better awareness about confusing which waypoint is the initial approach fix.  Something as simple as giving the IAF; D25; and the D11 fixes names would make the charts and radio communications less confusing.  Also; communicating to the crews that this approach clearance can be easily misunderstood.
A320 First Officer reported descending early on the VOR-DME-2 Runway 16 approach to MMSD; citing chart confusion and poor ATC communications as contributing.