Seems like the article wants you to accept a very narrow definition of recycling. reusing perfectly good shoes fits the definition of recycling in my book.
The phrase is "reduce, reuse, recycle" because it's intended to be an ordered taxonomy: reduction comes first, following by reusing existing materials, followed by recycling them into new materials.
Reusing perfectly good shoes is a good outcome, and nobody in this thread has suggested otherwise. But it isn't recycling in the way that Dow stated, either literally (since they claimed a polar opposite use) or definitionally (since the shoes aren't turned into new shoes).
It's not considered recycling because reuse is explicitly better than recycling. You aren't supposed to equate the two because it would imply recycling was a viable alternative to reuse.
Reusing perfectly good shoes is a good outcome, and nobody in this thread has suggested otherwise. But it isn't recycling in the way that Dow stated, either literally (since they claimed a polar opposite use) or definitionally (since the shoes aren't turned into new shoes).