As with most things in Thailand that may have been the case for some obscure reason but certainly not the norm and I can't imagine that the DLT would tell someone to apply at the police station
Immigration or the Police do not issue driving licences, the local DLT does but you'll also need a medical certificate from a doctor or clinic and I assume no residence certificate as you have a yellow book.
The 60 day Tourist Visa is not free, the current 60 day Visa Exempt entry stamp is not a visa and is free, try sticking to the correct terminology if you want to help not confuse people seeking information ;)