EMB-505 flight crew reported receiving a low altitude alert on approach and confusion regarding an approach clearance.

Short flight from ZZZ1 to ZZZ. Both airports were departing/landing to the North. There was a non-convective weather cell with precipitation, just West of both airports. 100 OVC, 10 SM in the general area, with reduced visibility near the precipitation cell. We departed Runway XXL at ZZZ1 with a right downwind vector towards ZZZ. Departure took place after sunrise, all terrain was visible and identifiable. 6 NM from ZZZ we reported field in sight and were cleared visual approach XYL via right downwind. Leveling at 4000 pattern altitude on downwind, the Captain and I decided that due to surrounding terrain, a right pattern may not be doable without triggering warnings and requested vectors for the RNAV visual XYL. ATC advised that we were under MVA, issued a low altitude alert, and we climbed uneventfully to the assigned altitude. Few minutes later, ATC cleared us for the "RNAV XYL approach", told us that ZZZ Tower just opened and handed us off. On TWR/CTAF frequency, ATC told us that ZZZ Tower will be closed for another 4.5 minutes, but continued to communicate with us. Shortly after, Tower Controller advised us "for informational purposes" that approach told them that we were cleared RNAV-B approach, but were instead flying RNAV visual XYL (different lateral track). We advised Tower Controller that we actually requested the RNAV visual, which we in fact did, and continued to uneventful landing XYL. On our landing rollout, ZZZ Tower opened, and class D airspace went into effect. It appears that both us and Approach Controller experienced expectation bias with respect to the approach clearance, and ultimately miscommunicated. As a flight crew, we briefed and expected to fly the RNAV visual XYL inbound, but never ruled out the traffic pattern, despite high terrain nearby. When ATC cleared us for the visual, right traffic, we followed that instruction, even though that was the inferior plan. Per ZZZ [informational safety bulletin], most of our Terrain Avoidance Warning System (TAWS) events occur when attempting to maneuver in the pattern, in close proximity to surrounding terrain. I suggest including instructions into ZZZ, pages recommending a straight in approach via the RNAV visual, and discouraging the right downwind Runway XY, due to potentially unstable approach and a TAWS event triggered by the 4174 ft. obstacle. Additionally, I suggest part-time ATCT Controllers refrain from providing instructions to/communicating with aircraft on CTAF, as it created some confusion regarding whether the Tower was already open and if we should expect ATC services. Regarding the approach clearance miscommunication, we should have queried the Controller for the full clearance phraseology, as "Cleared RNAV XYL" was ambiguous. Lastly, it was a XA00 LCL show to ferry the airplane to ZZZ into position for next morning departure, we were undoubtedly less alert due to operating during the window of circadian low. If we positioned the aircraft the evening prior, we could have avoided that.