Date: 202110
Local time of day: 1201-1800
Location: ZZZZ.ARTCC, FO
Flight conditions: VMC
Light: Daylight
Aircraft: B737-800
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Descent
Airspace: Class B ZZZZ
ATC / advisory: Center ZZZZ
Reporter organization: Air Carrier
Reporter function: First Officer; Pilot Flying
Reporter qualification: Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Human factors: Situational Awareness; Human-Machine Interface; Confusion; Communication Breakdown
Second reporter function: Captain; Pilot Not Flying
Anomaly: ATC Issue All Types; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter CFTT / CFIT
Detector: Person Air Traffic Control; Person Flight Crew
When detected: In-flight
Result: Flight Crew Took Evasive Action
Contributing factors: Human Factors; Procedure
Primary problem: Procedure

Narrative: ZZZZZ 1B arrival into ZZZZZ. Received clearance 'descent to FL250 at ZZZZZ.' Complied. After ZZZZZ; received clearance 'descend via ZZZZZ 1B; maintain 14 thousand;' We started our VNAV path descend. Passing ZZZZ1; we were given vectors; heading 070 while still descending to 14K. To the northwest of ZZZZ2; we were given clearance 'direct to ZZZZ3 and descent to 10 thousand (or descend and maintain 10 thousand; definitely not DESCEND VIA'). LNAV and VNAVing to ZZZZ3; noticed we are not descending below 14K due to the altitude restrictions in the FMS; and knowing we weren't given descend via for the 10K; I either V/S or LVL CHGed to start descending to 10K. Passing 11.5K; VFR; it seemed odd that they would want us that much lower than published. We asked approach to confirm descend to 10K. ATC responded with we're below the altitude on the arrival; I immediately dialed in something higher in the altitude window and started a climb. At the same time we received CAUTION TERRAIN; ATC came back and said we're given a descend via to 10K and now to climb and maintain 12 thousand. We climbed back up to 12K and completed the arrival and approach. Being a domestic pilot my entire time at [Company]; pretty much all of it in ZZZ. This might be the second time I've been to ZZZZ with the previous occasion probably over 6 years ago. Only ever flew into [Country] a handful of times and no [Continent]. I made sure I read all the airport familiarization pages; all the Jepp pages; all the [Country] info in the FOM. And was extremely vigilant regarding 'direct to vs direct' and 'descend via and descend to' clearances. Those were the key words I was listening for. Me and the Captain both agree and the second clearance did not include descend via. I do believe my lack of outside of US experience played a role. I trusted foreign ATC and thought the that low descent was common local procedure. In training and all the distance learning modules; we get hammered on the difference between DIRECT TO and DIRECT. I don't recall as much of an emphasis on DESCEND VIA vs DESCEND UNRESTRICTED (which is official ICAO phraseology on an arrival) and DESCEND TO/DESCEND AND MAINTAIN; which is a FAA phraseology for US arrivals meaning to descend unrestricted. I believe these points should be emphasized in future training modules. In regards to ZZZZ ATC; in my opinion they should only use 2 descent terminologies on arrivals with altitude restrictions. Either DESCEND VIA or DESCEND UNRESTRICTED. Using a DESCEND AND MAINTAIN or DESCEND TO clearance on an arrival with altitude restrictions; from what I can tell is non-standard per our FOM and should not be used.


Second narrative: CLEARANCE......Descending Via ZZZZZ 1B except maintain 14;000. Just after ZZZZZ vectored off the STAR to the east for spacing and given a descent to 10;000 ft. with multiple speed changes. Approaching 10;000 ft. in VMC conditions I queried the Controller about our altitude he said we should be at 14;000 ft. descending via. We were on vectors. We immediately initiated a climb to 14;000 ft. and as per his request leveled at 12;000 ft. We were cleared direct to ZZZZ3 and were told to rejoin the STAR. We landed XXL I believe there was a freq change to a new controller who forgot about us. He made no mention of our altitude until I asked him to confirm. There was also no mention of rejoining the STAR.Remain on the STAR/SID to the max extent possible.


Synopsis: B737-800 flight crew reported communications issues resulted in a descent below assigned altitude during the approach. Flight crew climbed up to correct altitude and continued the approach to landing.