Generally speaking, most people who run do not use all the features of a high-end running watch. An exercise watch that has a simple chronograph will work just fine. A chronograph is simply a timer. You press a button to start the timer and you press the same or a different button to stop the timer. At the end, the exercise watch will display the number of minutes and seconds that have elapsed.
A fitness watch is the same as an exercise watch. At its core, it contains a chronograph timer. You can use the term fitness watch or exercise watch to mean a watch for running.
Most fitness and exercise watches are digital. Digital quartz will get you the most accurate time. But, unless you are running a timed 100 meter dash, accuaracy that is off within a second is fine.