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

Narrative

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.