Military UAS Visual Observer reported a taxiway incursion while operating a ground vehicle resulting from a miscommunication with ATC about the taxi route.

Narrative

On day 0; I was a visual crewmember for Aircraft X; a Group X UAS conducting a test and evaluation flight at ZZZ.At approx XA:30; I contacted Ground to request transit for the landing aircraft. Upon reaching Tower; I requested 'Tower; Site X request.' with the airfield diagram and a digital airfield chart in operation. When Tower responded; I requested transit to Taxiway 1 as 'Site X; request transit via Taxiway 2; Taxiway 1; Cross XX for the north side of Taxiway 1'To the best of my memory; Tower read back 'Taxiway 2; Taxiway 1; cleared across XX for Taxiway 1'. At this point; I actuated my flashers and proceeded onto Site X; traveling NE on Taxiway 2 taxiway and approached Taxiway 1 and Taxiway 2 and Rwy XX.Assuming I had clearance; I cleared the approach; crosswind and final of Rwy XX visually and completed the crossing uneventfully. Upon clearing the runway at Taxiway 1 I conducted a U-turn to establish the vehicle at the Taxiway 1 runup area.After approx 5 minutes; Control inquired if I completed the Rwy XX crossing. I confirmed that I did and was at Taxiway 1 run up. Ground then asked if I transited Taxiway 2-Taxiway 3-Taxiway 1 to the run up. As I did not receive Taxiway 3 clearance with the readback; I told the controller the route I described; which I believe was Taxiway 2-Taxiway 1-Cross XX-North Taxiway 1. Post our conversation; an Airfield Management vehicle driven by Person A and one other individual approached the vehicle; where l produced an electronic copy of my airfield DL and reiterated my route using the vehicle's Airfield Diagram. I then proceeded to execute the aircraft X landing and taxi to Site X at XB:54.I believe the confusion occurred during the initial taxi request. The most preferred way would have been to allow Ground to prescribe the transit path; vs the driver requesting one. In lieu of this; during my request; I should have reiterated that I would ultimately like to cross runway XX. It is normal request for visual observers in vehicles to transit through the Rwy XX underrun; and I believe I may have said that when I wanted to actually cross at the threshold which caused the confusion. I should have also re-iterated crossing all runways; especially an active one. Conducting briefs on runway incursions is something our Pilot cadre does frequently; but special emphasis on the expected path or nuances of the underrun has already briefed to the aircrew.