What's the best aerobic exercise that will help me stay cut but still grow in strength?

Swimming. If you're talking about aerobic exercise that doesn't involve equipment or fitness classes. I'd definitely go with swimming. It's easy on the joints and works all of your muscles. It won't make them big and bulky though-- it makes them lean and long, so it depends on the look you're going for. Overall, I can't think of any other exercise that has the same benefits. Have you noticed that after swimming most people are feel quite hungry? It's because it works your whole body really hard, even though it isn't hard on the body and feels great. If more people switched to swimming rather than jogged, there'd be less injuries and joint damage in the world. Even die-hard runners will admit that it's hard on the joints and will probably cause problems for them later in life.

On the surface, rowing or sprint intervals would be a good answer. But the truth is, any cardio exercise will help you do that. You just have to manage how much cardio you do, clean up your diet, and hit the weights hard. Also, you may want to define your goal. Are you trying to gain lean mass or are you trying to improve your definition? It is possible to do both at the same time but it is very difficult. Studies have shown that it is best to focus on one or the either to get best results.