Cook ware is the number one item to look at. Aluminum pots/pans will cause a greying to your sauce if they are not clad in stainless steel. You'll get a reaction with acids and the aluminum that will cause disclouration to a light coloured sauce.
Aluminum gives great heat conduction for the dollar but is reactive. That's why you'll see so many present day aluminum pans will be aluminum cored with a steel outer to stop that reaction but give the rapid heat transfer.
As you see from the variety of advise from reputable sources, many combinations of hot/cold roux and liquid will work.
From a convenience point of view, you want at least one of them hot in order to speed the integration. If you started both of them cold, it would probably work but take a while to warm up to melt the butter in the roux, and free the flour particles to integrate and create the sauce@mdash;and it will won't fully thicken until the mixture is at a boil, so this is inefficient. You would probably also have to mash and stir more frequently to prevent local burning.
If you start with both hot, it will also work, although you might have to whisk relatively quickly—but who has both a hot base and a hot roux sitting around? Usually, at least one was prepared in advance, even if only by a few minutes. Batali is recommending this method, but note that in the professional kitchen, where he lives, time is the most constrained resource, so this is his method of choice. His line cooks probably have hot roux ready at all times.
A hot roux is easier to scoop and measure.
A hot liquid will quickly melt and dissolve the roux.
But in any case, if the roux is properly made, the starch granules are surrounded by fat, and so are not going to cause lumping in any case, despite Ramsay's advise. As soon as the mixture is warm enough to melt the butter, the particles are going to move away from one another, and won't be able to clump together before they are hydrated.
So I would not worry about this in the slightest.
Best Answer
After spending more time on search engines and thanks to the comment from belisarius...
Medium refers to the thickness level of the final sauce, and is controlled by the ratio of roux (butter/flour) to milk. For 1 cup of milk:
Sources: 1, 2