Accuacy implies precision, but not conversely: A statement can be precise without being accurate, but it cannot truly be accurate without being precise.
But I acknowledge, albeit reluctantly, that some people use the words as you say.
I would say, "Trenton is on the planet Earth (despite the evidence of our senses)," is accurate, but not precise. While saying "the North east corner of Trenton is a Latitude 10 (little round thing) 30' 45" S, Longitude 45 (little round thing) 21' 30" E." is precise as all get out, but woah is it inaccurate.
But, as I say, YMMV.
You mean there's nothing like a slide fit with statements? If I tell you someone is an thin as a rail, it could be an accurate statement, yet it lacks precision.