Air carrier pilot reported a track heading deviation due to miscommunication with NCT departure about a vector to the next fix.

Narrative

Aircraft X SJC-ZZZ received flight plan from Dispatch with the Loupe5 departure Runway 30R; received CPDLC clearance with a change to Techy 3 RNAV reprogrammed FMC prior to pushback reviewed entire departure. We departed following the Techy 3. Turned southbound on the climb with SJC Departure giving the climb unrestricted to 19;000 feet after SPTNS at 5;000 feet restriction; then Controller gave us a right 340 heading as we started the turn the Controller gave us direct ILC when able. Put direct ILC into FMC on autopilot the aircraft went left instead of doing a full 270 right turn which Controller gave us prior. Which is what the aircraft was programmed to do. In our left turn to direct ILC the Controller came back and said they wanted a right turn to 340 then direct ILC. Both myself and SIC (Second in Command) looked at each other and said that was not how we understood the clearance. We returned to heading select and went right to 340 once reaching 340 then went direct ILC. ILC was about a 030 we were on the departure on a heading of 125.The Controller should of gave us a heading of 340 let us turn to 340 then gave us direct upon rolling out not as we were still on the 125 heading. The automation will take the aircraft to the closest direct which in this case is a left turn.