THE MAN IN THE TRENCH COAT and THE BURGLAR are not, I would say, definite descriptions, they are incomplete description. Some argues that every description is incomplete. If you want to analyze this description in Russell'a way [= an a existential quantifier] then you should have properly restricted domain of quatification. When you analyze this sentence you will get something like ExEy S(where E stands for existential quatifier and P for an atom sentence). Were you quantifying over full domain (whole world) then THE BURGLAR isn't satified by exactly one object, it's not what we want.