I understand the process because I have lived in Thailand for over a decade but tourists do not, that is why they keep asking. The government and the media should not have used the June 1 date.
it is strange for Thailand to announce 60 day visa exempt, but then have no idea when they will implement. It is understandable why people planning their holiday would want to know.
1. From your profile, you are from the UK. UK passport holders receive a 30 day visa exempt on arrival. This is not a visa on arrival.
2. You can convert a visa exempt to a 3 month Non O based on being 50 or above and not working in Thailand as long as you can meet the requirements. In the last month of your 3 month Non O, you can apply for a one year extension.
This must be done at an Immigration office within Thailand, it cannot be done online.