This can be confusing for people so I could see how a person could do research and still not know the facts. Its important to note that many international airlines do not allow rear facing car seats, which means no infant seats. Sometimes people are flying on an international airline and actually don't know. What matters is who the flight is operated by rather than who you bought your ticket from. (When you buy your ticket, for example from Expedia, it will say "operated by blah blah" on your itinerary, this isn't secret info.) If you are flying a US carrier they are required to follow FAA rules and the FAA says any seat that is approved for use in aircraft by them can be used on a flight. Plenty of infant carriers, in fact possibly all currently on the market, in the US are approved for use in flight. Again, this is different in other countries.
Like Wendy said, even flight attendants are confused about the rules here and sometimes they give out false information, but there are documents you can bring with you from the FAA to help in your discussions with the flight attendants and I've found that once they checked their facts they have always agreed with what I posted above.
However, we are currently having a discussion on this board (and there are a few other places on the internet people are talking about this) about some customer service reps from United who claim that United is no longer following FAA policy on this point. We are in the process of researching this further.