How did 'under influence', as in in driving under influence, get equated to 'intoxicated'?
"driving under the influence of alcohol" - avoids there having to be a definition of drunk and people being able to argue in court about how drunk they were.
Somebody once escaped because the police claimed he was drunk. They or their lawyer argued that the police weren't medically qualified to asses if somebody was drunk.
Nowadays in most countries there is a simple blood alcohol level at which you are guilty, and a machine to determine this - but lawyers like tradition
I'm from the UK, and for several decades we've had a specific law against driving with more than a specified concentration of alcohol in the blood. Depending on various factors, some people would be 'stinking drunk' just below that level, whereas others might be stone-cold sober just above it.
But before that the law, it was already illegal to drive while incapable through intoxication. The law wouldn't have specified which intoxicants, obviously. In common parlance it's just driving under the influence [of intoxicants].
I can't imagine US law would fail to address the same issues, so let's be clear that we're not just talking about alcohol here. I've no doubt there have been prosecutions in the UK of people people who drove while woozy from prescription painkillers. Cars are lethal weapons, and drivers have a legal responsibility to ensure they're fit to operate them (that includes wearing glasses if appropriate).
The shortened form Driving under influence, which I personally have never actually heard here in the UK, probably isn't because of any tendency for casual speech to drop insignificant words like 'the'. It's most likely just an over-literal expansion of the standard US police acronym DUI for the offense. Which sadly they still have to refer to practically every day.