All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Mentioning sins is only forbidden if it is for the purpose of making it public and the like, as for mentioning the sin for a preponderant benefit, such as seeking a way to repentance and the like, or seeking pardon from the person whom you have wronged, then there is no harm in this. What is included in this is the example that you mentioned; informing of your lie, correcting the information, and withdrawing what you have informed about previously, then all this is part of your complete repentance. This is not from publicizing the sin, nor is it a forbidden mention of it.
Allah knows best.