lifter
The lifters are solid lift and do not have a hydraulic plunger.
The reason the Miata ticks IMOP is the filter is not in a up right position and when you start up the engine there is air sent to the lifters for a moment. Some engines that don't have a oil filter in a vertical position have a drain back valve in the filter. My turbo diesel Toyota factory oil filter did and it would pressure up the oil galley faster when you start the engine.
I have taken the valve covers off and started the engine and carefully placed my thumb on each rocker arm to see if I could feel the ticking noise. Some times you can take a long screw driver and use it to listen for noises.
My first guess is the fuel pump. I have seen the Chevy gas engines push rods ball ends plating go bad and cause noise, or worn rocker. Since the Cummings is different try to isolate the noise. We had a welder in the shop that was knocking and it was bought new last year, pulling the rocker covers it was found to be a valve sticking.
I have seen flex plates make noises that would sound like a rod bearing. lol jimk