The way I was taught it (and confirmed by this answer and the comment on it), the Hebrew word "tirtzach" refers to murder, and is distinguished from the H-R-G and M-T roots which mean "kill." Therefore, the pasuk which reads "lo tirtzach" (do not murder) is not in conflict with judicial killing (tehargeno, or yumat).
However in the case of a killer who has to run to an ir miklat, the go'el hadam who catches up and kills the unintentional killer, first, is said to kill (M-T) him and then is said to R-TZ-CH him (all from B'midbar 35): יט גֹּאֵל הַדָּם, הוּא יָמִית אֶת-הָרֹצֵחַ:
כז וּמָצָא אֹתוֹ, גֹּאֵל הַדָּם, מִחוּץ, לִגְבוּל עִיר מִקְלָטוֹ: וְרָצַח גֹּאֵל הַדָּם, אֶת-הָרֹצֵחַ--אֵין לוֹ, דָּם
I know that in the second pasuk, the situation has changed (the accused has left the safety of an ir miklat) but if his killing of the accused is "extra judicial" then he would be held liable. The end of the pasuk says he isn't.
A following pasuk is even more confusing: ל כָּל-מַכֵּה-נֶפֶשׁ--לְפִי עֵדִים, יִרְצַח אֶת-הָרֹצֵחַ
No context is given, but the killing of someone deemed textually a "murderer" is called "murder."
The Torah Temima quotes the gemara in Makot 12a which posits opinions that it is a mitzvah to kill the accused if he leaves the city (only in particular situations, granted) so the title "murder" should not apply.
Am I missing something in understanding the root R-TZ-CH?