The US Embassy is oversimplifying things.
Under US law you have to apply for a visa if you have been arrested (even if not convicted) for certain categories of crime.
The Embassy does not trust the UK public to decide if their offence falls into those categories - probably rightly as it's fairly technical. So they advise everyone who has been arrested to get a visa, just to play safe.
If you can sign the declaration on the visa waiver application form that you have never been arrested or convicted of a crime involving 'moral turpitude' (and a few others) then you can legally go without a visa. If you can't then you should apply for one.
OTH hand it's often stated here that if you just sign the form anyway they've no way of telling that you are lying.