There seems to be a lot of confusion in this thread.

1. Given that OP is traveling on a USA Passport, a Visa is not required, neither for transit, nor to enter the UK.
2. An Electronic Travel Authorisation (ETA) is not a visa, it is what its name says: an authorization to travel. It is the UK equivalent of the US ESTA. That ETAs and ESTAs generally only apply to visa-exempt populations does not make them a visa.
3. As pointed out upthread, an ETA is not strictly required for most transits through LHR (though it was originally). However, ETAs are cheap. easy to get. and valid for 2 years. Besides avoiding any confusion with Delta at check-in, it will also give you the option to go landside at LHR to transit, or if your connection is delayed or canceled.
4. When transferring from T3 to T2, if you follow the purple Flight Connections signs, you will end up on an airside bus that takes you from T3 to T2. Technically you are going outside, but you will not clear immigration (because you are not entering the UK) or customs (again you are not entering the UK. You will clear security when you arrive at T2.
   1. @andrewk829: The reason for the security check at T2 is that not all arrivals are from "clean" countries, and so you are not considered sterile on arrival. While there are limited experiments for "clean" transfers, those only apply to select flights from DFW and ATL at the moment.
   2. If you do not follow the Flight Connections side, you will end up at immigration, and you will enter the UK. As a US passport holder you can use the e-gates, which are generally pretty fast. If you do this, you can simply walk from T3 to T2 via the underground passageway (past the Tube station) and then reclear security when you get to T2. This may or may not be faster than taking the Flight Connections bus.

tl;dr: It would be penny-wise but pound-foolish to not get the ETA simply to save ~$20.