You are almost correct. The incorrect part is that if the soft fork chain has more hash rate, the old clients will have to discard their bip110 non-copmlient blocks and they will accept the aoft form chain. That is exactly because it is a soft fork - blocks mined by the new rules are seen as valid by the old clients.
The soft forking chain gains majority hashrate and wipes out the non-forking chain. In that case, the miners that mined the non-forking chain bet incorrectly and lose their rewards.
You are almost correct. The incorrect part is that if the soft fork chain has more hash rate, the old clients will have to discard their bip110 non-copmlient blocks and they will accept the aoft form chain. That is exactly because it is a soft fork - blocks mined by the new rules are seen as valid by the old clients.