It depends - how long are you planning to stay this time?
In theory, you don't have to apply in advance because you can be given a 6-month stamp at the border (like when you visited for 10 days)... however, there's no guarantee that you will be allowed into the UK and given a stamp if you want to stay for the whole 6 months.
The longer you try to stay in one trip, the more scrutiny you will be under when you arrive at the border, and the more likely you will be refused entry and sent straight back to the US.
A 2-week trip won't cause suspicion as many people can and do visit the UK for a 2-week vacation. However, not many people can afford to take 6 months off work to be a tourist. So, the immigration officer at the border will need to be sure you are a genuine tourist and that you aren't trying to use the visitor visa to essentially live in the UK.
The longer your trip, the more evidence you'll need to show of ties to the US which mean you must return to the US within 6 months. You'll need to be able to show:
- enough money to support yourself for 6 months without working
- a job/home/strong ties to return to in the US which means you won't be tempted to try to stay in the UK.
If you are worried about being allowed in for a full 6 months at the border, then it might be an idea to apply for a visitor visa in advance, so that you will already have confirmation that you have been approved to enter as a visitor, and if it does happen to be refused, you will only have lost the cost of the visa (about £90) and you won't have wasted time or money on flights.
If you apply in advance you should not book any flights until you have the visa in your hands. But if you 'apply' at the border you will need a return ticket.
See here for how to apply for a visitor visa in advance:
https://www.gov.uk/standard-visitor-visa/overviewSent from my iPhone using Tapatalk