I think the general principle is that defence lawyers aren't expected to contradict their clients' probably false testimony, and are perfectly entitled (encouraged, even) to deliver a "my client maintains the shot was fired by someone else, and nobody witnessed the shot" defence whilst being pretty convinced they fired the shot, but if they close with a series of factual claims about "Richie Bottoms" whilst knowing full well that name was made up on the spot by their client they could be in trouble. Doing their job properly means trying to persuade their client to stick to technicalities and avoid the "Richie Bottoms" defence because it'll fall apart, not patching the holes in it.

But the fact we're debating how open defence lawyers are supposed to be and there is jurisdictional variation is a good indication of why criminals might feel they're best served lying to their defence lawyer, and lying as elaborately as possible rather than just avoiding confessing. The potential consequences of telling the truth are much worse than merely having your public defender think you're a fool. Plus if you're intending to test whether your denials are believable before you use them in court, you won't get a better opportunity than a privilege-protected conversation with someone who's seen a lot of excuses...