Is it considered a mistake or hypocrisy to show the text of a conversation to friends to change their wrong perception of my intentions, taking into account the importance of their opinion of me and my eagerness to clarify my position?

There is no harm for a Muslim in dispelling ill-will directed at them. If what you intend to present to them to clarify a misunderstanding does not involve anything prohibited, then it is permissible to present it. It is important to pay attention to the necessity of adhering to the Shariah guidelines when dealing with non-mahram men. It is not permissible to mix with them or to be in seclusion with them without a need.