Truthfully, we use neither now that we have a manure spreader (an old Milbrook model for $100)which we hook up to the tractor and drive right down the barn aisle way or drive right into the run in sheds. This is really the way to go if you have the equipment! WE just shovel right into the spreader. But knowing that it took me almost 25 years before I got one - I prefer a good contractor's wheelbarrow - I wouldn't use the one pictured because if you need to stop, "muck" can fall out as you have to lay it on the ground. The biggest problem with a wheelbarrow if making sure you load your manure in so the load is balanced so you don't tip over.
I have used the muck carts but don't like using them because you need to lift the bucket out to pour out the contents and if you don't have a bad back you can develop one pretty quickly with this method especially if you are also cleaning big horses' stalls too. That said I do love the cart or a garden cart with 4 wheels for moving hay, shavings and putting my full and empty feed buckets in to make the rounds.
Another thing we also use is to get a heavy duty wheelbarrow that we got at a tag sale. We removed the barrow part (then I spray painted it and fill it with dirt and flowers for a small enclosed garden) and then my husband screwed wood fence slats (2x1s work also) onto a frame (looks like a deck chair without legs or arms)= he has also used an old manager to make the"chair" and then screwed this onto the wheel part of the wheelbarrow (easiest if the wheelbarrow has a wood not metal frame). Now I can easily load 4 - 50 lb bales of hay and easily push it down the alley way or even out to dry lots )or winter pastures) without having to make too many trips back for hay as we have 26 horses.
Edited to say that the "chair" is facing whoever is pushing to hold the hay from slipping off the end. Wgat I really want now is a bobcat!