There isn't a fixed dollar figure. A luxury flip means buying and reselling near the top of the local price range, where the buyer pool is small and buyers expect finishes most flippers never touch. In one metro that starts around $900k. In another it starts at $4M. What makes it luxury flipping is the small buyer pool and the finish standard, and every cost scales with the price too.
The loose usage is the one you found. Listing agents call anything above the local median a luxury renovation opportunity because it reads well in copy. If a house would be bought by an ordinary move-up family with an ordinary mortgage, you're doing a mainstream flip at a higher price, whatever the ad says.
Something that surprises people coming from paper: pricing evidence gets thin fast at the top. A mainstream flip might have twenty comparable sales inside six months. A $3M house might have three, and roughly a dozen states don't make sale prices public at all, so depending where you are your resale estimate may rest on agent-reported figures or MLS records rather than recorded prices. That widens the range your number can land in, and at these prices a 5 percent miss is six figures.
Carrying cost is the other half of it. A mainstream flip sitting two extra months is annoying. A luxury flip sitting eight extra months can take the whole profit, which is why this strategy belongs to people with a lot of spare cash.