What visa details have to be included in the airline records?

For airlines, the passenger name record (PNR) database has basic information (name, ticketing, itinerary) but can also include information required by many countries (e.g., date of birth, gender, passport details).

Prior to boarding, airlines use the IATA Timatic database system to verify that passengers have the required documents (e.g., passport, visa, health) for their destination and any transit point(s).

The governments of the United States, United Kingdom, and Canada require that all airlines collect Advanced Passenger Information (API/APIS) from all passengers prior to travel (within 72 hours prior to departure if booked in advance, or at time of booking if made less than 72 hours prior).

APIS asks for all names as written in your passport, gender, date of birth, nationality, country of residence; passport number, country of issue, and expiry date. You are not required to provide any other documentary information.

First of all, it is absolutely irrelevant what details you put into check in (or ticket purchase) forms, outside of your name. Nobody cares about them and airlines just throw away whatever you entered during check-in when they swipe your passport at the gate.

However for flights outbound to the US you can make your life a tiny bit easier by entering the correct details of whatever passport contains your US visa. This would allow you to check in online rather than checking in at the airport, although that's not a big time saver if you have checked in luggage and selected your seat in advance. On the flight back to India you can enter whatever you want to the check in form and it won't affect anything as India doesn't require airlines to verify visas before online check-in.

Enjoy your flight :)