Something to consider?
I like Carl's Rubbermaid Cart (see link, p.3) though I have never used one, because it will contain all the bark, dirt, melting snow, etc... that inevitably falls off of wood and, in an open cart, would fall on the floor of your home. So, too, obviously, would a wheel barrow, which I use in my cottage--I wheel it up a snowy board, one step up, and right inside, next to the stove--the dog approves of the style, apparently, and my g.f. avoids the place, so no complaints.
For our main house, however, (complete with three-step porch) I need something a little more stable and wooden-floor-friendly than my metal-legged wheelbarrow.
I like Eric's open-style cart (or Goose's HF cart) because you can pull the wood off of the cart, as you work your way down, as opposed to bending and "picking" the wood from inside the Rubber Maid Cart's tub.
So I would ask Carl: do you find that it's a pain to pick the wood out of the bottom of the cart, or am I just incredibly lazy? The bottom of that tub looks lower than my wheel barrow's bottom, and the open carts seem easier to unpack than a 'barrow or a tub. Since I don't like to clean, either, your Rubbermaid Cart appeals to me, as I say, because at least it's containing all of the wood's associated "effluvia," to be dumped outside, upon the reload.
Just thought it was worth mentioning the "lifting vs. cleaning" trade off.
Maybe a hybrid, two-wheeled cart like the Rubbermaid, but with open sides and just a "bellypan," (to catch the detritus) is the ideal solution?