That is correct,
For the indentification law, any Thai goverment issued ID card should be sifficient.
But there can is also the requirement to proof your legal sttaus in the country. (Visa check)
Here is the issue, it is not sure if this can be done with a copy or does it have the original passport, Depending on their mode they say one or the other.
Also officially only a immigration officer can check your legal status as normal police officers don't have access to the immigration information.