This root was the basis for “sword” words throughout Tolkien’s life, but the meaning of the root itself shifted over time. The first appearance of this root was as ᴱ√MAKA in the Qenya Lexicon of the 1910s, unglossed but with derivatives like ᴱQ. mak- “slay”, ᴱQ. makil “sword”, and ᴱQ. makka “slaughter” (QL/57-58). The root was also given the gloss “slay” in a section of the contemporaneous Gnomish Lexicon along with etymologies of names of the various Valar, but this section was deleted (GL/18). Derivatives of this root appeared elsewhere in the Gnomish Lexicon, however, such as G. mactha- “slay, kill” and G. magli “a great sword” (GL/55). Thus, the meaning of this root in the earliest period seems to be “slay”.
The root ᴹ√MAK appeared in The Etymologies of the 1930s, but the entry went through quite a few revisions. The gloss apparently was “cleave” >> “kill, cleave with sword” >> “sword, or verbal [stem] fight with sword, cleave” >> “sword, or as verb-stem: fight (with sword), cleave” (EtyAC/MAK; Ety/MAK). Thus the 1930s root was more directly connected with swords and wielding swords, and its derivatives included ᴹQ. makil/N. magol “sword” and ᴹQ. mahta-/N. maetha- “fight” (Ety/MAK).
In Tolkien’s later writings, the words for “sword” remained nearly the same: Q. macil and S. megil (PE17/130, 147), but the gloss of the root √MAK varied considerably based on whatever linguistic puzzle Tolkien was trying to solve at that particular moment. In notes associated with the Quendi and Eldar essay from 1959-60, its gloss was very similar to that from The Etymologies: “cut, hew with a sharp edge” (VT39/11). In notes associated with The Shibboleth of Fëanor from 1968 Tolkien glossed {m(b)aka- >>} maka- as “forge metal” as part of a new etymology of the name S. Maglor as an adaptation Q. Makalaure “Forging Gold” (PM/353; VT41/10). In notes on Eldarin Hands, Fingers and Numerals from the late 1960s Tolkien gave √MAK “strike” as the basis for ✶makwā > S. mâb “hand”, but this idea was rejected immediately (VT47/19). This was part of Tolkien’s rather surprising decision to abandon the long-standing root √MAP (VT47/20 note #13); elsewhere in these notes he said √MAK meant “kill, slay” as it did in the 1910s (VT47/20).
Lokyt suggested in a Discord chat from 2018 that there may be a common underlying meaning for all these glosses, referring to “the movement one does when chopping with a tool”. Assuming this is true, the other associations of the root (“sword; slay; fight; forge”) may be the result of a narrowing of the meaning of the root in more specific contexts. While it is hard to know whether Tolkien himself interpreted the root this way, I think this is the best way to treat the root for purposes of Neo-Eldarin, as it allows us to retain the largest set of derivatives of the root.
This root has a long history in Tolkien’s writing, but it evolved in meaning over Tolkien’s life. The earliest form of this root was ᴱ√KILI “edge” from the Qenya Lexicon of the 1910s, with derivatives like ᴱQ. kíla/G. cilm “edge, rim” and ᴱQ. kilme/G. cail “lip” (QL/46, GL/24, 26). In The Etymologies of the 1930s it became ᴹ√KIL “divide” with derivatives ᴹQ. kilya/N. cîl “cleft, pass between hills, gorge” (Ety/KIL). This sense seems to have been abandoned when Tolkien changed ᴹQ. Kalakilya “Pass of Light” (LR/173) to Q. Calacirya (LotR/377).
In etymological notes from 1969 Tolkien gave √KIL “choose, select” vs. √KIM “edge, brink of”, along with various derivatives of √KIL having to do with choices (PE22/149). One of these, Q. cilmë, appeared as an element in words from several essays on Elvish names from the 1950s and 60s: Q. Essecilmë “[Ceremony of] Name-choosing” (MR/214; PE21/84) and Q. kilmessi “self-names, names of personal choice” (PM/339). But it seems even in the 1969 note mentioned above that Tolkien was considering the earlier senses of the root, since he first gave √KIL/√KILIM as the basis for edge words (PE22/149 note #26).
The difference in sense is not that great between 1910s ᴱ√KILI “edge” >> 1930s ᴹ√KIL “separate” >> 1960s √KIL “choose”, so Tolkien’s conceptual shifts on the meaning of this root seem to be gradual, with him still considering the earlier meanings into the late 1960s.