Hi Heidi
Im not sure about more recent models but my Mums 2005 LG machine has a rinse and spin cycle on the dial which does a low level cottons rinse and upto 1200rpm spin cycle, If one selects "Medic Rinse" with this cycle it does a full 4 Rinses and spin cycle for cottons.
For more delicate than cotton cycles you have to select the programme for the fabric and select a cold wash and let the machine do its full thing.
I cant think why they would omit a single rinse and spin, best bet would be to select a spin cycle and see what Time comes up in the display, note down what it is, then select a cotton cycle and see if pressing any combinations of buttons reduces the time of the whole wash to say 5-6 minutes more than the spin cycle alone. Id be sure this was a rinse and spin then.
Failing that I reckon you would have to just select 30 minute Quickwash and select Cold as the temperature,
If it has a 15 minute quickwash perhaps they omitted the rinse and spin and this cycle could be used as so. On Mums machine the spin cycle selected automatically on synthetics or quickwash is a default 800rpm however by pressing the spin speed button one can then reselect 1200rpm if one needs a faster final speed,
Hope this helps,
Rob