Yes, I'm sure that is indeed the law.
However, having lived and worked in the middle of Bangkok for 25+ years I've never heard of anyone in trouble for only producing a photocopy of a license. In fact I've been stopped and searched whilst going home in a taxi several times (usually towards the end of the month when officer's rents are due) and no problem.
I even vaguely recall a few years ago, a police chief (probably Thonglor) published a poster warning against pickpockets and advising carrying a photocopy of passport. It could even have been at Songkran time.
What a pity the police don't do something similar now when many maybe seeking the help of police.
I should also add, Peter, that I NEVER carry important original documents with me except when needed, like to Chiang Wattana etc.