Your logic at the end is correct, high inventory turnover and lower days of inventory on hand mean your cash operating cycle will be shorter.
The cash operating cycle measures the gap between paying your suppliers and collecting cash from your customers.That gap needs financing. The longer it is, the more it costs you to finance it. To best manage your cash you want it to be as short as possible (or even negative - collect from customers before you pay your suppliers).
The time it takes from buying your raw materials to collecting from your customers can be measured by adding together days of inventory on hand (how long it takes for your raw materials to be turned into a final product and sold) and receivables days (how many days once the final product is sold is it before you collect the cash from your customers).
The good news for most businesses is that they don’t pay for their raw materials immediately. So days of payables (how long you take to pay your suppliers after buying your raw materials) reduces the gap between the payment and receipt from customers.
The operating cycle is therefore DOH + DOS - DOP
Reducing days of inventory on hand, which means increasing inventory turnover (how quickly inventory is sold), therefore reduces the operating cycle.