Recent studies on a quasi-one-dimensional (quasi-1D) charge-density-wave (CDW) system, In atomic wires on Si(111), have brought intriguing issues on topological solitons in quasi-1D systems: the existence of few-atom-sized solitons, chirality of solitons, and the realization of logic exploiting the chirality switching. Using scanning tunneling microscopy and first-principles calculations, we show that the previously reported "short" phase-flip defects are In adatoms and thus have nonsolitonic nature, resolving the controversy over the existence of highly localized solitons. The observed "long" phase-flip and phase-slip defects are genuine solitons with and without chirality, respectively. While achiral solitons (phase-slip defects) can exist on the pristine CDW (8×2) surface, chiral solitons (phase-flip defects) cannot due to their breakage of 8×2 ordering. The chiral solitons can exist only when they are trapped by In adatoms and constitute a part of a closed-loop domain wall. The intertwinement of chiral solitons and In adatoms implies the limitations of the previously proposed logic utilizing soliton chirality, but it provides an opportunity to realize this by controlling the In-adatom defect.