Nine different entries in the OED for high. Goodness. But as far as I can tell, it is all the same word high which JRRT uses, meaning both far from bottom to top and elevated in social stature — even “5.002 for he could feel inside that it was high time for some meal or other”. High is a Germanic word and its earlier origins are debated and if you want to wander down a rabbit hole of Tocharian A and Tocharian B and their friends, do spend a pleasant hour with this word’s etymology!

• 1.004 about half our height,
• 1.079 The pines were roaring on the height,
• 1.109 “Five feet high the door
• 2.028 rising higher
• 2.028 and higher,
• 3.035 very old swords of the High Elves of the West,
• 3.040 beside the plain runes which say ‘five feet high the door
• 4.003 in the high hope of a midsummer morning,
• 4.005 They were high up
• 5.002 for he could feel inside that it was high time for some meal or other;
• 5.035 “But in low place/ Not in high place.'”
• 5.055 And beats high mountain down.
• 6.003 out of the little high valley,
• 6.034 And we are still pretty high up.
• 6.052 and up they went as high as ever they could trust the branches.
• 6.070 It was high summer,
• 6.080 as he got ready to spring down from on high
• 6.084 above the highest branches.
• 6.085 and they were high up
• 7.002 and saw that the birds were already high up
• 7.005 even from their great height,
• 7.009 except high
• 7.029 and beyond these to a high thorn-hedge
• 7.031 high
• 7.055 We were crossing by the High Pass
• 7.106 a splash of white on the floor came from the high moon,
• 7.130 As soon as they left his high hedges
• 7.151 high in the East,
• 8.028 and as high as he could throw it.
• 8.029 High it sprang
• 8.041 and so he must be light enough for the highest
• 8.088 in a row from a high branch.
• 8.131 They differed from the High Elves of the West,
• 8.132 at the feet of the high wooded lands.
• 9.015 a very high opinion indeed)
• 10.006 over high waterfalls
• 10.009 and towed away round the high shoulder of rock
• 10.034 and high over the lake.
• 10.036 Even Bilbo was given a seat at the high table,
• 11.005 and ever higher above them.
• 11.005 in a height called Ravenhill.
• 11.013 somewhere high above the cliff at the valley’s head
• 11.017 while Bofur was hauled up to the higher camp.
• 11.019 that led higher
• 11.019 and higher on to the mountain;
• 11.032 who had been watching from a high perch
• 11.038 A door five feet high
• 12.007 at the height of their wealth
• 13.044 when suddenly the roof sprang high
• 13.064 and climbed the high bank.
• 14.002 Only its high peak could they see
• 14.013 the dragon’s wrath blazed to its height,
• 14.014 He circled for a while high
• 14.015 Flames unquenchable sprang high into the night.
• 14.023 and crashed down from on high
• 14.025 The waxing moon rose higher
• 14.025 and higher
• 14.043 higher up the shore;
• 15.010 This very height was once named Ravenhill,
• 15.028 But now their hopes were higher;
• 15.028 and high,
• 17.031 The dwarves are exceedingly strong for their height,
• 17.048 climbed to the height of the Eastern shoulder
• 17.056 Rocks were hurled down from on high
• 18.052 in high summer.
• 18.053 But they came to that high point at morning,
• 18.053 On its highest peak