When it comes to the obligation to tell the truth, there isn't all that much difference between a lawyer's obligations and those of a pro se litigant -- at least as far objective truth.
But not every question has a single truthful answer. Professionalism rules impose some higher standards on lawyers in cases that are a bit murkier than just asking, "Were you at the Capitol on January 6?"
In the United States, the analogous rule lays out some bright-line rules.
Rule 3.3: Candor Toward the Tribunal
(a) A lawyer shall not knowingly:
make a false statement of fact or law to a tribunal or fail to correct a false statement of material fact or law previously made to the tribunal by the lawyer;
fail to disclose to the tribunal legal authority in the controlling jurisdiction known to the lawyer to be directly adverse to the position of the client and not disclosed by opposing counsel; or
offer evidence that the lawyer knows to be false. If a lawyer, the lawyer’s client, or a witness called by the lawyer, has offered material evidence and the lawyer comes to know of its falsity, the lawyer shall take reasonable remedial measures, including, if necessary, disclosure to the tribunal. A lawyer may refuse to offer evidence, other than the testimony of a defendant in a criminal matter, that the lawyer reasonably believes is false.
(b) A lawyer who represents a client in an adjudicative proceeding and who knows that a person intends to engage, is engaging or has engaged in criminal or fraudulent conduct related to the proceeding shall take reasonable remedial measures, including, if necessary, disclosure to the tribunal.
(c) The duties stated in paragraphs (a) and (b) continue to the conclusion of the proceeding, and apply even if compliance requires disclosure of information otherwise protected by Rule 1.6.
(d) In an ex parte proceeding, a lawyer shall inform the tribunal of all material facts known to the lawyer that will enable the tribunal to make an informed decision, whether or not the facts are adverse.
Subsection (a)(1) gives a good example of where the duties of honesty diverge for lawyers and pro se parties. If a plaintiff tells the court honestly -- but mistakenly -- that he lost $1 million in profits, but later discovers that he only lost $100,000, his lawyer has a clear obligation to correct that statement for the court; the pro se plaintiff's obligation is not clear.
Similarly, if a plaintiff tells the court that he is entitled to those lost profits if he can prove elements A, B, C, and D, but later learns that the Supreme Court has also imposed a requirement that he prove E, the lawyer has an obligation to notify the court of this development; the pro se plaintiff does not.
On the "overriding" language: I don't read it as generally having any effect on a lawyer's duty to be honest to his client. Instead, it means that the lawyer's duty to the court overrides the lawyer's duty to the client.
In either of the above hypotheticals, for example, the lawyer acted honestly and ethically in presenting his evidence and argument, even though his statements turned out to be false. Correcting the record on either point would reduce or possibly eliminate his client's likely recovery and be against his client's best interests.
Once the lawyer discovers the error, he is therefore faced with a conflict of interest: he has a duty to act in his client's best interest, but he also has a duty of honesty to the court. Chapter 13 says that his duty to the court overrides his duty to his client.