I've done both. Honestly I have more luck cutting carbs at weight loss, (it's really rapid), but I wonder how much weight lost is fat vs muscle and water. Eliminating carbs eventually makes my kidneys hurt, so I think cutting calories, (plus exercise) is much safer.
Recently neither, I've cut out sugar and most carbs.
I count calories.
Neither one, I just weigh myself every day and if I gained weight the day before it means I ate too much.