I'd go with 4gb and the 1030. That combo will let you play lots of games, not all games, well. Stay away from the 8gb and up games.
Meanwhile, integrated graphics won't let you play much of anything. You'd have to stick to old, if not 2d, games.
If you can live with integrated graphics for now and know you won't have long until you can afford a better videocard, that's the only scenario where the integrated graphics makes sense.