Fathom has the REO origin exactly right. Adding the strict line for the second half.
Pre-foreclosure describes a stage in time. The owner has defaulted and a formal notice has been recorded, but the auction hasn't happened yet. The owner still holds title and can still sell. That's all the word means. It says nothing about how much is owed or what the house is worth.
Short sale describes a price relationship. It's a sale where the agreed price is less than the total owed on the loans, so the lender has to agree in writing to accept less than full payoff and release its lien anyway. The lender is a required party to the deal, and short sale approvals routinely take months because the file goes through a loss mitigation review, sometimes with a mortgage insurer also needing to sign off.
So a short sale is usually happening during pre-foreclosure, and that's why the words get blurred. But a pre-foreclosure owner with $200k of equity who sells for $250k isn't a short sale at all, it's an ordinary sale by someone in trouble. And a short sale can happen for someone who's current on payments but underwater and relocating. The two words answer different questions: when in the process, and whether the price covers the debt.
The part that catches people reading these contracts for the first time is that in a short sale the seller's signature doesn't close anything. Every deadline in that purchase agreement is effectively suspended until the lender's approval letter arrives, and the approval letter often comes back with its own terms, a different price, a cap on seller credits, a required closing date. If you're reviewing one of these, the approval letter is the document that actually governs, and how much power it has to override the signed contract varies by state. Worth having a local real estate attorney read the first one you touch.