ok you own a token because there is consensus you own it. Not because of math but because everyone believes (and agrees to acknowledge the truth of) the results of that math.
You create a contact between you and idex. It can only return a token to you or move your token forward with your signature.
Once inside contract (channel), internal order book stores your balance? No. That is all on chain still.
You place a sell order (matter). Internal order book then claims your currency? Or is already in your channel? Probably JIT claim. You sign to allow this to happen.
Order book shows your order. Matched via what (price? timing?) And then….
New contract created to send ETH/XYZ. Buyer signs to execute the transfer. Order book is updated to pending. Eth contact fires. Once mined and settled, orderbook receives event and updates to processed.