Answer: Staphylococcus aureus.
Step 1: The key word is the very short 3-hour incubation. Staphylococcal food poisoning is caused by a PREFORMED heat-stable enterotoxin already present in the food, so symptoms hit fast, usually within 30 minutes to 6 hours.
Step 2: The picture fits: abrupt nausea, vomiting and cramps after a party meal (often custards, dairy, mayonnaise handled by a colonised food handler). Illness is self-limiting and gone within a day.
Step 3: The others incubate longer. Salmonella and Clostridium perfringens take 8 to 72 hours because they need bacterial multiplication or in-vivo toxin action, and Clostridium botulinum (12 to 36 hours) causes descending paralysis, not early vomiting.
Ref: CDC and Jawetz Medical Microbiology.