They all seem to deal with some part of a process of reconciliation but could someone explain the difference between Teshuvah (תשובה) Charatah (חרטה) and L'hiNachem (להינחם) ?
While the terms you use may seem related to the same central concept, they are actually opposites.
This is especially true for Teshuvah(Repentance) and Charatah(Remorse)
Regret is rooted in both of these concepts as you need regret to embody either one.
Teshuvah is a fantastic word because at the core of the concept is the idea that every Jew is ultimately good. The concept is tied to the idea that we all start off with a divine beginning and that over time we simply move away from the intended path.
Teshuvah speaks of returning to your pure beginning. The act of repenting for the wickedness isn't to start fresh as a new person but to instead return to your original nature. The central belief in Judaism is that every Jew has a pure nature at their core. We are all inherently good people who simply fall from the divine path we were meant to walk. Teshuvah implies that a person is returning to this natural state rather than destroying their past mistakes and starting anew. This is a different philosophy from Charatah.
Charatah is remorse for the past and an intention to start anew. This idea runs in contrast to Teshuvah in that you intend to completely reinvent yourself rather than return to an original nature. As mentioned previously, the main difference between Teshuvah and Charatah is that the individual is overlooking their inherently pure nature. Instead of making peace with your past mistakes and returning to this "true" you that existed prior, you seek to destroy that history of past mistakes and reinvent yourself anew.
Both of these concepts are rooted in L'hiNachem(regret) as a person cannot enter into either without having the inner realization that they have been living the wrong type of life.
So the main difference between these two concepts is how the individual is looking upon themselves. Am I ultimately good at my core and can return to my pure past or do I believe I need to destroy my past to truly be good?