I'm using an RX100 Mk II and I can confirm you that it has always a pre-flash...even in manual mode! I would suspect that it is the same for your MkIII.
I considered buying a YS-D2 to replace my YS-90DX but I finally choose an INON Z240 Type 4, so from my reading of the YS-D2 manual, my understanding is that DS-TTL II is the new name for the DS-TTL(probably new features were added on from the DS-TTL mentioned in the YS-01 manual). These are not two different modes on your strobe.
If I were you I would follow Hellhole advice and go Manual. Set your strobe on M (with the two lightning bolts, indicating pre-flash) and your RX100 MkIII on manual. With a bit of experimentation with the settings (both on your flash ad strobe) you will achieve the exposure you are seeking.