I traveled through the US on 13 August to get to India. I didn't leave the airport, just got on a connecting flight, but the immigration officer stamped my passport to leave the US by February. Do I have to pass back through the US before February 12, although I didn't stay in the US. I am a Jamaican passport holder.
The stamp in your passport means that you must leave the US on or before 12th February. You did that – you left later on 13th August – so you have nothing to worry about. On your return journey to Jamaica, you'll get another stamp that will also say "You must leave within six months."
The reason for the date on the stamp is that, in the US, there's no separate system for transit. Once you'd been through passport control, there's nothing to stop you picking up your luggage and leaving through the airport's front door, instead of taking your flight to India. So the US treats transit through the country in mostly the same way as visiting it.
No, you have used that entry and left the US long ago.
Next time you transit through US you will receive a new entry. You don't have to transit within the period provided by the last one.
You can easily confirm your departure record with the CBP here. It will have your entry and exit record related to the stamp in question. When you visit/transit next time and if they admit you, it will be a new entry and a subsequent exit with its own conditions.
Because you already hold a visa superior to the transit visa they allowed you to stay for 6 months, and you left the US within 6 hours (hypothetically). No one is going to take any issue with that.
Next time you transit through the USA might as well enjoy the privileges granted to you due to your visa and the subsequent admission instead of staying put at the airport for God knows how long.
You may need to keep evidence of leaving, or arrival at your next port out of transit. It is not unheard of to be falsely accused of overstaying a visa based on flaky records. You should be OK, but it never hurts to be sure.