Terra, the different steering levels note the bounds on the flow pattern for a particular internsity threshold. So, when it says 400mb-850mb, it means that they are taking the average of the winds in the level of the atmosphere between 850mb (roughly 1500m above the surface) and 400mb (roughly 8000m above the surface). The same holds true for the different levels, just with different bounds.

Different levels are selected for various storm intensities because stronger storms tend to be deeper, meaning that their circulations extend higher up in the atmosphere. Thus, either a larger difference between the bounds (like 250-850) or overall higher (in the atmosphere) values of the bounds, such as 200-700, are better indicators for these stronger storms, more often steered by mid- and upper-level troughs and ridges. Conversely, weaker systems tend to be steered more by the low level flow, so regimes like the 700-850mb or 450-850mb layer that take in to account low and mid level flow are more useful.

Of course, as with any product, the values are a little arbitrary and not all storms will fit perfectly within one regime. And, the winds will change over time, so it is not an end-all tool for predicting storm motion. But, one of the NHC's favorite ways to depict storm motion after the fact -- and perhaps during the storm, though I'm not certain of this -- is to take a mean 1000mb-100mb wind as the steering flow. Essentially, you capture everything from the surface to the top of the troposphere (the region in which we live) with this. It is attempts to capture portions of this that have also been proven useful in tracking storms, such as the products you linked to (which I really like myself).