You are correct in stating that government service (aka. civil service) pensions are covered as an exception in the Thai-foreign country DTA's. You are correct in stating that state pensions are not mentioned. You are, I assume, correct in stating both are taxable income in Australia.
However, under Thai tax legislation, your Australian state pension is not taxable, it is tax exempt.
What does this mean? I will talk about the UK because that is what I know.
Your UK state pension will be included in your taxable income when calculating your UK tax bill. Even though UK citizens may be non-resident in the UK for tax purposes, they still suffer tax on their UK income.
If they have to or choose to fill in a Thai tax return as they are resident in Thailand for tax purposes, they should omit their UK State Pension (if they remitted it to Thailand) as it is exempt income in Thailand.
I assume a similar position applies for Australia.