This describes auto tires, and is basically the same as the Continental animation you've provided:
http://www.goodyeartires.com/about/diversity/how_built.html
OK, so I'm responding to a internet posting and didn't feel like wasting everyone's time trying to show off how much I imagine myself to know. You are right that the components are
assembled inside-out for the vast majority of tires. That gets all the bits into their proper order and place, because you can't open and close the mold for each layer.This takes us through Step 5.
Then the tires are placed into the mold, and heated to fuse the layers, belts, and tread together, mold the treads and sidewall markings, and complete the vulcanization of the rubber. This is step 6 in the process, and the mold I was speaking of, and shown very well in the animation you've provided. This is representative of the way almost all tires are made. I described it as tread being "first layer on" because the heat of the mold fuses things together from the outside of the tire inward. This forces the tire into the semi-permanent "U" cross-section that we are all familiar with.
Open-built tires proceed somewhat differently. The assembly is similar through step 5, though with slight differences - components are pre-vulcanized where possible, and the tread strip will already have it's tread embossed on it if applicable. Then (and in intermediate steps in some tires) the tires are hand vulcanized (that is, the gluing rubber holding the pre-vulcanized parts together is vulcanized) right on the building drum, without ever seeing the inside of a mold. By argument, this results in a more supple tire as the molding pressures work themselves out as the tire is made - somewhat analgous to the process of stress-relieving spokes when building a wheel. Whether you buy that it's better or merely more traditional is where the marketing fun comes in.
The difference is usually easy to spot. Conventionally made tires have the mold sprues sticking out around them like an impotent porcupine, and a telltale mold break line running around the middle of the tread. Open-built tires do not show these sprues or break lines. Some conventionally-made tires don't either, as the manufacturer takes the extra step to clean them off before shipping, but that is relatively rare. Typically, open-built tires lay and fold flatter when off the rim than conventional tires, though the gap has narrowed in recent years.