Trail runner here, if running in the desert is considered a trail. Depending on location, its advised to have some sort of pack. If you at your urban wilderness park and have good cell coverage on a short run, then just make sure you have some fluids with you.

For longer distances or rural/state park stuff, I advise at least a fanny pack of preparedness gear if you plan on holding a water bottle otherwise a hydration pack should be used. Again with gear in case you hurt or the weather changes.

There is no difference between a trail run and a hike, except location and how much gear is on your back.