Dear British Guy:
Perhaps I can clear this up for you.
The US Immigration Laws permit a short course of flight training such as obtaining an FAA license on a B-1/B-2. If you are a British Passport holder you can enter the United States on the Visa Waiver program (which does not require a visa) and complete your short course of training and get your license. The B-1/B-2 usually allows a period of stay in the United States of six months renewable in the United States for a further six months. The Visa Waiver allows a maximum period of stay in the United States of 90 days.
If you intend remaining in the United States for a long period of time and you are engaging in a program of formal study then the M-1 or J-1 visa are options to consider.
If you are a green card holder, you are a permanent resident of the United States and the above does not apply. You cannot then enter on a non-immigrant visa unless you intend to abandon or have already abandoned your permanent resident status.
When entering the United States hand your passport to the Immigration Officer. If he asks you a question answer the question directly and honestly. The principal issue on entry is if you are going to leave the United States at the end of your visit. If the Immigration Officer asks you the purpose of your trip it is a visit. If asked if you are going to fly state that you are. There is nothing wrong with this.
Flight clearances regarding aircraft of 12,500lbs or more is a separate issue to the above If you intend training or flying an aircraft in the United States that weighs more than 12,500lbs then get precleared before you enter the US. Once you are precleared the above applies.
Enjoy your flying in the US.