There's a great deal of opinion on this subject, the majority of the testing I've seen favors the multiple flip method. My own tests agree, you get a better result from more flipping.
Type of meat has no influence on the method, although it impacts how long you cook it. For example a rump steak is a bit denser than a sirloin or ribeye so it takes longer to cook.
The reason that steaks cook unevenly when flipped once is that the top of the steak heats as you cook the bottom, when you flip it what was the top has a warmer starting point so that side ends up hotter at the end of cooking. You can get evenness from the one flip method by doing 2/3 of the cooking time on the first side and 1/3 on the other side (that's approximate from my own testing). The benefit of a single flip is it's less time spent on the steak, if you have a lot going on in the kitchen you have more time to spend on other tasks.
Other than that multiple flips are the way to go as you get even cooking, less curling (although you can reduce fat curling by cutting across the fat every inch, this gets a better result in general), and the steak tends to cook faster. Note this is for thicker steaks, for thin steaks you really just want to get a crust on as quickly as you can, so a single flip makes sense.
I've never noticed a difference in juiciness with either method, other aspects of technique are more important, for instance with a thick steak you want to cook it mostly on the cool side of the barbecue and then crisp it up on high heat as opposed to cooking it on high heat the whole time.