The manual for my Harrison L5 lathe advises setting the cabinet on a slight slope down to the drain-hole. I have considered fitting a suds system but to be honest I am not sure it's justified for the amount of use of this lathe. Besides, the way the machine is bolted to the cabinet would probably ensure coolant leaking through onto the floor below – a point to consider with other lathes, too.
Using a plumbing trap (sink or bath type)? A bottle trap, possibly, and there is no reason it cannot be PVC, but don't use the version that is no more than a U-bend.
Remember that the trap designed for the kitchen or bathroom is not a sediment-collector. It is a water-seal to prevent drain miasmas from creeping up into the house.
The inverted strainer over the chip-tray drain is the best bet and often fitted as standard. One might be made from the stainless-steel mesh sold for smoke-box spark arrestors.
Then the weir in the coolant tank; and that could be an open-topped box easily lifted out for cleaning,