Tolkien used this root for “lie (down)” for most of his life. It appeared as ᴱ√KAYA “lie, rest; dwell” in the Qenya Lexicon of the 1910s (QL/46), but in that document there was another root ᴱ√KAMA “lie down”, with the derivative ᴱQ. kama- “to lie down” (QL/44). There is no sign of ᴱ√KAMA¹ being used this way after the 1910s, and in the 1920s Early Qenya Grammar, the verb for “lie down” was kaita- (PE14/58), which in the Qenya Lexicon of the 1910s had the transitive sense “to place” (QL/44).
The root ᴹ√KAY “lie down” reappeared in The Etymologies of the 1930s (Ety/KAY), and it appeared again in the Quenya Verbal System (QVS) from 1948 as the basis for the verb ᴹQ. kaita- “to lie (down)” (PE22/126). The root and associated Quenya verb continued to appear in Tolkien’s later writings all the way up through the late 1960s, always with the sense “lie” (PE17/72; PE22/156).
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:
1910s ᴱ√LATA “✱level, smooth” as alternate to ᴱ√LAHA “✱open” (QL/50-51).
1930s ᴹ√LAT “lie open” compared to (unglossed) ᴹ√LAD (Ety/LAD, LAT).
1940s ᴹ√LAT “be extended, stretch, be situated (of an area)” contrasted with ᴹ√LAD “lie flat, be flat” (PE22/126).
1959-60 √LAT “open, unenclosed, free to entry” contrasted with √PAK “shut” (VT41/5-6).
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.