You are discussing two different ideas. First it must be understood that the Days of Moshiach is a long period as found in Zohar 1:140a:51-52, over 100 years according to many. According to Sefer Avodat HaKodesh, Section 2, Chapter 38 there are stages to that period.
The initial period is that which is described by the phrase found in Brachot 34b, Sanhedrin 99a, Shabbat 151b, Pesachim 68a and Zohar 1:140a:50, that there is no difference between this world and the Days of Moshiach except for servitude to the Monarchy alone. According to some, that means the Jewish people will no longer be subjugated by the non-Jewish nations. According to the Ba'al Shem Tov in Keter Shem Tov, chapter 46a it means servitude to G-d's Kingdom, that Jews will be free to serve G-d without interference.
During that period, the natural world will still operate as we know it and at least the possibility for murder will exist. The Lubavitcher Rebbe emphasizes that because the three additional cities of refuge pertain to an actual mitzvah, they must be understood according to their plain sense and not some allegorical view.
When Moshiach, meaning someone presumed to be Moshiach is present, there will be a process of refinement in the world eventually returning things to the state of being that existed prior to the sin of the Tree of Knowledge. Some of the details about that refining period are enumerated in the Mishnah Torah, Laws of Kings and their Wars, chapter 11 in regard to someone presumed to be Moshiach.
The second period discussed in Avodat HaKodesh is the period when miraculous things like people having lifespans that extend to 1000 years like a tree (Man is like a tree in the field, Devarim 20:19, Isaiah 65:22), the elimination of death and the return of the miraculous, curative fruit of the land of Israel called Gluska'ote, like is found in Ketubot 111b, will take place. These are like the miraculous fruit described in the episode of the spies in the Torah (BaMidbar 13:25-26).
Some view these as miraculous events, while others, like Rabbi Meir Ibn Gabbai in Sefer Avodat HaKodesh, emphasize that according to the Torah, these things were the natural order before the sin of the Tree of Knowledge. So in reality, we are only returning to how things were intended at the beginning of creation.
This period is also when when resurrection of the dead will begin for the Select Individuals (Yechidei Segulah) like is mentioned in Sukkah 52b quoting Micah 5:4 and Zechariah 2:3. It is the period that in Mishnah Torah cited above in chapter 12 is associated with King Moshiach, meaning Moshiach with certainty like is understood from Yoma 5b and the discussion of Moshe and Aharon being involved when the Temple service is reinstituted. The conclusion of this period is when the general resurrection will take place.
There are varying opinions about whether these miraculous events will be limited to within the borders of the land of Israel of will occur outside the land of Israel. According to some, this is resolved through the idea that ultimately the borders of the land of Israel will be expanded to encompass the whole world.
There are also varying views about whether this elimination of death will only pertain to Jews or if it will ultimately include the non-Jewish world too. If one considers the language of BaMidbar 35:15 discussing the concept of the cities of refuge, it specifies that these cities are for the children of Israel, and for the Gerim, and for the Resident Aliens. So it could be emphasizing the view that the Days of Moshiach will be different for Jews and non-Jews.