What are the rules for UK tourists regarding visa exemptions and entry limits to Thailand?

Although there’s no offical limits other than the 2 land borders, be wary about entering via the 2 major airports in particular if you have spent over 180 days in country in the past 12 months. The immigration officers at the Bangkok airports in particular are at their discretion pulling aside people who have been in country over 180 days as a regular tourist. It happened to me on 1st May and I needed approval by their superior to enter. If you are over that limit then you really need to look at other options to extending a visit ie making use of the 2 land border bounces (with a border run company) if not yet used, or applying for METV well in advance of a long planned trip.

I've done 3 times with no problems. I think if you are going back to UK or USA (for me) for a few months than coming back to Thailand, you should be OK. It's the ones doing one day or no days border bounces to neighbouring countries, who may have problems.

went 9 times last year on visa exempt, but normally 10 to 14 days at a time

Get a Visa.

Best site for precise information is your nearest Thai Embassy and where you can apply.

Previously have been 3 times per year with no problems.

Went 4 times in one year 30 days at a time

There is no legal limit to how many times you can arrive visa exempt by air. The only written limit is that you can enter 2 times per calendar year by land border without a visa.

The Thai embassy in the UK does not make the immigration policies, so regardless of what they put on their website, you can enter as many times as they will allow you.