Do I need a visa to transit between German cities when I have a U.S. visa?

Sorry, you'll need a Schengen visa. The lines you are quoting are about transiting via a single airport ("MUC or FRA", not "and"), meaning you fly in, change planes and fly out without ever entering Germany proper. But you want to take a domestic flight, which does require going through immigration and hence a valid visa that lets you enter Germany.

There is no contradiction, the Lufthansa website you found should be read as meaning transiting through either Munich or Frankfurt but not both at the same time (an exclusive "or" or XOR if you are familiar with logic or programming).

You will need a Schengen uniform visa (and not merely an airport transit visa) for this route.

The other two answers have already pointed out that you need a permission to enter the Schengen area, but they didn’t go into much detail why.

As was noted, the central leg is a fully domestic, intra-Schengen one, while the first and last legs are extra-Schengen international ones. (domestic and intra-Schengen can be read as synonyms in the Schengen area; when non-Europeans think international they mean something that to European airports is extra-Schengen.)

If you arrive in either Munich of Frankfurt on an extra-Schengen flight, you will arrive in a relatively segregated part of the airport and before accessing your luggage or anything, you will have to pass entry immigration. Since these two airports are the largest in Germany and since they have a high percentage and number of connecting travellers, provisions have been made to simplify transiting between extra-Schengen flights. I’m not entirely sure on the full procedure since I never transferred from extra-Schengen to extra-Schengen on either of those airports. But I can imagine separate walkways bringing you from the international arrivals more or less directly to the international departures without having to come close to the domestic part of the airport.

Of course, nothing would be stopping you from attempting to go somewhere else. But ‘somewhere else’ in the extra-Schengen departure area is limited to other flights away from Schengen; to actually enter the country you would need to pass through a proper immigration control at some point. So extra-Schengen to extra-Schengen is a ‘low risk’ itinerary.

Your central leg is domestic, though. It does not depart from the extra-Schengen departures but from the intra-Schengen (domestic) departure area. This is not separated from the general public area by any further immigration controls; only by airside security screening. Maybe it still is hard to get from the domestic departure area back out to the landside part of the airport; I never tried it.

However, your arrival in Munich will also be on the domestic arrival side. The only thing separating domestic arrivals from the general public area is a simple gate at the end of the luggage carousels; no immigration control nobody checking that you’re going the right way. This way, it would be extremely easy to say ‘yes, I’m just transiting; I’ll board my plane bound for India in Munich’ while actually exiting the airport, forfeiting the final leg and entering Germany illegally.

And that is why you cannot just use an airside transit visa (or none at all because Munich/Frankfurt airport) but you need a full-fledged Schengen visa to properly enter the Schengen area. They want to know who they’re letting in and it’s a whole different level of scrutiny compared to airside transit.