Walking the cup is used when running uphill usuallyYou can do a weave by either walking the cup or by what you refer to as "standard" TIG welding. . .
. . .usually (not always) the purpose of a weave is to make a wider pass when working with thicker materials (but sometimes it can be for cosmetic reasons as well); you normally try to match the fillet to the thickness of the material and that is not always possible in a single pass. This method is used a lot in pipe welding, but has found it's way into our world as well.
There are a lot of videos of this method out there. . .doing a YouTube search should turn up a more than you have time to watch. There are a few that have been posted on this forum as well. . .I don't have a link to any ready to go, but a search should serve you well.