Precision positioning plays critical roles in 5G-Advanced and 6G systems for various indoor and outdoor applications. Conventionally, positioning accuracy can be dramatically degraded under non-line-of-sight channels or with moving terminal nodes (TNs). To overcome this issue, we propose an orthogonal time-frequency space (OTFS) based positioning with applying distributed cooperative positioning (DCoP) at the ends of TNs, which can yield an ultra-most precise positioning. This is so, since the positioning with OTFS is robust against channel delays and Doppler spreads, and superior to conventional orthogonal frequency-division multiplexing (OFDM) based positioning. Further, DCoP uses cooperations such that TNs can serve as additional anchor nodes (ANs) to others, and improves the positioning accuracy for all. With this proposal, we further elaborate the positioning-reference-signal with OTFS (PRS-OTFS) in delay-Doppler (DD) domain, and design an OTFS-transceiver that is backward compatible to OFDM. Furthermore, we also propose a DD-domain time-of-arrival estimation algorithm that exploits the OTFS waveform and outperforms conventional time-frequency based estimators. Moreover, we propose a loopy belief-propagation algorithm associated with belief-discretizing for an effective implementation and message exchange in DCoP. In addition, we show that the average Cramer-Rao low-bound per TN can decrease faster than linearly in the number of cooperating TNs. We illustrate with simulations that the proposed OTFS-DCoP scheme is very effective in improving positioning accuracy within a few iterations, and robust against double-selective channels.