This root has a lengthy history in Tolkien’s development of the Elvish languages.
Its development seems to have begun in the Qenya Lexicon of the 1910s with unglossed ᴱ√LAHA which had derivatives like ᴱQ. lá “moor, heath, open space” and ᴱQ. lāta- “spread, extend, lie (of country)” (QL/50). This root was compared to a variety of other roots: ᴱ√ALA, ᴱ√AŘA, ᴱ√ṆŘṆ, ᴱ√LAŘA and ᴱ√LATA, but the last of these was a later addition, both to the list and as an entry in the lexicon (QL/51). This newer entry had derivatives like ᴱQ. latwa “smooth, glossy” and ᴱQ. latsin(a) “level, smooth”, the latter transferred from ᴱ√LAHA. This new root also had Gnomish derivatives such as G. lad “a level, a flat; fair dealing” and G. ladin “level, smooth; fair, equitable” (GL/52).
After the 1910s it seems ᴱ√LAHA “✱open” faded out of use (though it may have remerged much later as √LAƷ “cross, pass over, go beyond”: PE17/91), but ᴱ√LATA “✱level, smooth” reappeared in The Etymologies of the 1930s as ᴹ√LAT “lie open” with derivatives like ᴹQ. latin(a) “open, free, cleared (of land)” and N. lhand “open space, level” (Ety/LAT). It is also evident in N. -lad “plain” from names like N. Lithlad “Plain of Ash” and N. Dagorlad “Battle Plain” from Lord of the Rings drafts (TI/208, 389), both of which reappeared in the published version (LotR/243, 636).
The root appeared again in the Quenya Verbal System (QVS) from 1948 with the gloss “be extended, stretch, be situated (of an area)” and a Quenya verb ᴹQ. lat- of similar meaning (PE22/126). In this document it was compared to ᴹ√LAD “lie flat, be flat”, and Tolkien said the two roots were confused in Noldorin as a way of explaining N. laden “flat (and wide)”.
The next appearance of √LAT is in etymological notes from 1959-60 with the gloss “open, unenclosed, free to entry” and contrasted with √PAK “shut” (VT41/5-6). In these notes the derivative Q. latina also reappeared with a meaning similar to the one it had in The Etymologies: “free (of movement), not encumbered with obstacles”. The last appearance of √LAT in currently published materials is in 1968 notes on D/L variations in Common Eldarin, where it was glossed “at ground level, low; lowlying” and compared to √DAT “fall down” (VT48/24); in this document it had no derivatives.
All this amounts to considerable conceptual variation in the meaning of this root. To summarize:
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1910s ᴱ√LATA “✱level, smooth” as alternate to ᴱ√LAHA “✱open” (QL/50-51).
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1930s ᴹ√LAT “lie open” compared to (unglossed) ᴹ√LAD (Ety/LAD, LAT).
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1940s ᴹ√LAT “be extended, stretch, be situated (of an area)” contrasted with ᴹ√LAD “lie flat, be flat” (PE22/126).
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1959-60 √LAT “open, unenclosed, free to entry” contrasted with √PAK “shut” (VT41/5-6).
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1968 √LAT “at ground level, low; lowlying” contrasted with √DAT “fall down” (VT48/24).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to assume this root has the sense “(lie) open, be extended, unenclosed”, as this is consistent with the largest number of derivatives of the root.
Counterpart to the root √N(D)IL, this root first appeared in a marginal note from The Etymologies of the 1930s as ᴹ√NDUR or ᴹ√DUR “bow down, bend (down), obey, serve”, an elaboration of ᴹ√NDŪ “go down, sink, set (of Sun, etc.)” (Ety/NDŪ; EtyAC/NDŪ). In the Quenya Verbal System (QVS) from 1948, ᴹ√NDUR was glossed “grow, be dark”, no doubt in connection to N./S. dûr “dark” (PE22/103), but later this “dark” word was connected more directly to √NDU “under, down”. Indeed, in a 1955 letter to David Masson, Tolkien gave √(N)DUR an origin distinct from √N(D)U, basing it on an independent root √DUR “to show special interest in things” (PE17/152); there was also an unglossed and deleted root ᴹ√DUR in The Etymologies of the 1930s (EtyAC/DUR). However, in a 1967 letter to Mr. Rang, -(n)dur again meant “to serve” (Let/386).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to keep the connection between √NDUR “bow down = serve” and √NDU “descend”. As for the semantic distinctions between -(n)dil and -(n)dur, see the entry on the root √N(D)IL for details.