Public defenders are lawyers and take the same oath as every other attorney. In fact, they take it twice--once when sworn into the bar and once when sworn in as a PD. As to resources, virtually all PD offices are understaff and under funded but they do what they can. In many states, the court will have to fund some things like investigators and experts. With that said, however, for serious felonies, most people can't afford to pay an attorney to defend that type of case and will have a public defender or a contracted defense attorney represent them. There's nothing wrong with defense counsel identifying themselves as a public defender or the prosecutor referring to them as that. If they are private counsel, they can certainly correct the misstatement if they want to.