Basically, you only become an overstayer if you stay in the UK for more than 6 months continuously... without leaving the country, even once.
I.e. if you entered the UK on February 1st and you didn't leave AT ALL until after August 1st, you would become an illegal overstayer... as you would have been in the UK continuously for more than the 6 months you were allowed when you entered.
However, as long as you leave before the 6 months is up, you do not become an overstayer... so if you left on February 11th, you were only in the UK continuously for 10 days of those 6 months.
When you entered again on February 16th, you got another 6 months, and you stayed continuously for only 3 months of your 6 months.
When you enter again on May 20th, you will only be in the UK continuously for about 12 days of your 6 months.
However, the general rule of thumb is that you should spend at least the same amount of time OUTSIDE the UK as you did INSIDE the UK on your last trip. So, as you've just stayed for almost 3 months (Feb to May), you should really wait another 3 months before returning - so you ideally shouldn't come back until August anyway.
If you've just been here 3 months and then try to come back in again only a week later, you may find you get trouble at the border and the immigration officer might feel you are spending too much time in the UK (they can refuse you entry if they think you're spending too much time here).
Basically, each time you enter the UK, you 'apply' for another 6-month visitor visa. It's up to the immigration officer at the airport as to whether they allow you those 6 months or not - they can let you in for only 2 days if they decide to. Or if they think you've spent too much time in the UK in the last 12 months, they can refuse you entry and send you back to the US. This is not overstaying, it's a refusal of entry into the country.
There is no hard and fast 6 months in 12 months rule, and this is explicitly stated in the visitor visa rules:
Frequent or successive visits: how to assess if an applicant is making the UK their main home or place of work
You should check the applicant’s travel history: how long are they spending in the UK and how frequently are they returning? You must assess if they are, in effect, making the UK their main home.
You should look at:
• the purpose of the visit and intended length of stay stated
• the number of visits made over the past 12 months, including the length of stay on each occasion, the time elapsed since the last visit, and if this amounts to the individual spending more time in the UK than in their home country
• the purpose of return trips to the visitor’s home country and if this is used only to seek re-entry to the UK
• the links they have with their home country-consider especially any long term commitments and where the applicant is registered for tax purposes
• evidence the UK is their main place of residence, for example
o if they have registered with a general practitioner (GP)
o send their children to UK schools
• the history of previous applications, for example if the visitor has previously been refused under the family rules and subsequently wants to enter as a visitor you must assess if they are using the visitor route to avoid the rules in place for family migrants joining British or settled persons in the UK
There is no specified maximum period which an individual can spend in the UK in any period such as ‘6 months in 12 months’. However, if it is clear from an individual’s travel history that they are making the UK their home you should refuse their application
