You may also be asked for proof of your accommodation. Also it is not a 60 day visa or a visa of any kind. It is a 60 day Visa Exempt stamp that Immigration put in your passport.
You are correct. However your national licence, supported by an IDP is only valid for the 90 days while you are a tourist in Thailand. Once you have a long term visa of 90 days or longer or are on an extension of stay, then you are no longer a tourist and are required to have a valid Thai licence.
Firstly, you do not have a visa of any description. You have a "visa exempt" stamp in your passport which is not a visa. It sounds like you have entered Thailand many times using this method and want to stay lonmg term in the country. The visa exempt entry is intended for tourists and not for long term stay. At some point soon Immigration will deny you entry into Thailand. You need to get a long term visa from outside of Thailand related to the reason for your stay in the country.