Well, it seems the internet is full of fellow victims of this particular design tragedy, as the sources you've found are cries for help, not heroic solutions. Fear not! A rogue splash of 'Passion Pink' on your plush velvet throne is a crisis, but it's not an insurmountable one. Let's stage an intervention for your sofa.
First, a crucial word of warning. Velvet is the drama queen of fabrics and can be highly sensitive. Before you unleash any cleaning concoction, you absolutely must perform a spot test on a hidden area, like under a cushion or on a back seam. We're trying to save the sofa, not create a new, abstract art piece against your will.
If the polish is still wet, gently blot—do not rub!—with a paper towel to absorb the excess. If it's dry, you can try to gently scrape off the hardened polish with the edge of a credit card or a dull butter knife. Be delicate; you don't want to snag those luscious fibers.
Next, grab your weapon: a non-acetone nail polish remover or rubbing alcohol. Acetone is the mortal enemy of many fabrics and should be avoided like a bad first date. Apply a small amount of your chosen solvent to a clean, white cloth or a cotton swab. Gently dab at the stain, working from the outside in to prevent it from spreading. You'll see the polish begin to transfer to the cloth. Keep dabbing with a fresh part of the cloth until the color is gone.
Once the colorful offender has vanished, you need to clean up the cleaner. Mix a drop of mild dish soap with a bit of water and gently blot the area to remove any solvent residue. Follow this with a final blot using a cloth dampened with only plain water. Let the spot air dry completely. Don't be alarmed if the velvet looks a bit sad and flattened. Once it's bone dry, you can gently coax the pile back to its former glory with a very soft brush. And in the future, perhaps consider a manicure station with a slightly larger blast radius.
