You'll have decide for yourself what you "should" do and are "supposed to do". Different cultures exhibit different behaviors in different environments.
I keep all of my starters in the refrigerator. Some I use at least once per week, others I use every 2 weeks or so. They stay out of the refrigerator only during activation.
If you are using and feeding a starter frequently, it could stay out on the counter. The reason you put a starter in the refrigerator is to make it dormant so it doesn't need to be fed. However, if it is dormant long enough, it may take multiple feedings to revive. You can leave it dormant too long, so I make sure I feed any unused culture at least every month. YMMV.