Corporate pilot reported they triggered a low altitude warning from ATC during approach descent due to a misunderstanding of approach clearance instructions. Flight crew continued to landing.

Narrative

We conducted a flight from our home base in ZZZZ to ASE via a refueling stop in ZZZ.The weather at the time of the approach at XA53Z. Variable at 3 knots; 10 statute miles; Broken at 5000 ft. overcast at 7000ft;Temperature 3 Due Point 7 QNH 298. We were handed off from Denver Center to Aspen Approach and were vectored for the LOC-DME E for runway 15. Approaching waypoint JARGU we were advised by Aspen Approach that the preceding aircraft was visual with the airport over Waypoint KICER at 12;900 feet which is also what we experienced moments later. We were told to then contact the Tower and advise if wanted to continue visually. On contacting Aspen Tower I reported visual and we continued the approach maintaining our own terrain separation; only to be advised by the controller that we had triggered a Low Altitude Warning over Waypoint FIMSO and were asked again if we were visual which we confirmed. Upon landing and approaching the ramp we were asked to telephone the Tower on shutdown which we did only to discover that we had misunderstood instructions from the Approach/Tower controller as we had not formally requested a visual approach; mistakenly assuming our reporting of visual would be sufficient. Following our conversation with the Aspen controller we immediately contacted our Director of Aviation in ZZZ1 and also our Chief Pilot in ZZZZ.I have submitted a company Report to be read by all our flight crews and was also asked to submit this report. We intend to follow up on the points raised in this recent event at our next company Safety Meeting with our crews from the US and Europe.The Aircraft X crew are based in ZZZZ and operate mainly within Europe. A regular destination is ZZZZ1 which has similar mountainous terrain to Aspen. The initial Approach there is usually sequenced by controllers until the flight crew report visual when they are released to continue the approach maintaining their own terrain separation. Regrettably this crew did not fully appreciate the subtle differences in US terminology on transition from the Instrument Approach to the visual segment. These lessons now learnt will be discussed and there importance emphasized and circulated at our upcoming Safety meeting to insure this does not happen again.