Regarding your request for a definition of cool. Here is the "official" word from the OED (Note: some of the earlier quotes might not appear right because they are not translated to modern English letters). The first definition is the emotional state, the second refers to jazz, while the third is the "neat" definition.

4. a. Of persons (and their actions): Not heated by passion or emotion; unexcited, dispassionate; deliberate, not hasty; undisturbed, calm.

Beowulf 282 And þa cear wylmas colran wurdað. c1440 Chaucer's L.G.W. (MS. Gg. 4. 27) 258 Thow+thynkist in thyn wit that is ful cole That he nys but a verray propre fole That louyth paramouris to harde and hote. 1570 Levins Manip. 161 Coole, quietus. 1590 Shakes. Mids. N. v. i. 6 Such seething braines+that apprehend More than coole reason euer comprehends. 1611 Bible Prov. xvii. 27 A man of vnderstanding is of an excellent [marg. coole] spirit. 1716–8 Lady M. W. Montague Lett. xxii. 69 Upon cooler reflexion, I think I had done better to have left it alone. 1736 Butler Anal. ii. vii. 376 Some of them were men of the coolest tempers. 1798 Miller in Nicolas Disp. Nelson VII. clviii, I caused a cool and steady fire to be opened on them. 1848 Macaulay Hist. Eng. II. 491 The energy of the young prince had not then been found a match for the cool science of the veteran. 1855 Tennyson Maud ii. i, While she wept, and I strove to be cool.

d. Applied to jazz music: restrained or relaxed in style; also applied to the performer; opp. hot a. orig. U.S.

1947 (record by Charlie Parker Quartet, Dial 1015) Cool Blues. 1948 Life 11 Oct. 138 Bebop: New Jazz School is Led by Trumpeter Who is Hot, Cool and Gone. 1950 Christian Sci. Monitor 8 Feb. 15 Bop is ‘cool’ jazz in contrast to the ‘hot’ variety of the swing or Dixieland schools. 1953 Melody Maker 9 May 5 Hot and cool—you've got to hear the lot. 1955 L. Feather Encycl. Jazz (1956) 30 Cool jazz to most musicians and students denotes the understated, behind-the-beat style typified by the arrangements and soloists on the Davis records. 1957 H. Panassié in S. Traill Concerning Jazz 61 The ‘cool’ musicians+stopped using the traditional jazz technique and tone. 1962 J. Wain Strike Father Dead iv. 204 The new developments which were to become first bebop and then just bop and finally cool jazz.

e. Hence, characteristic of those who favour ‘cool’ music; relaxed; unemotional; also used loosely as a general term of approval; cool cat: see cat n.1 2c. colloq. (orig. U.S.).

1948 New Yorker 3 July 28 The bebop people have a language of their own.+ Their expressions of approval include ‘cool’! 1953 Time 14 Sept. 68/3 The latest Tin Pan Alley argot, where ‘cool’ means good, ‘crazy’ means wonderful. 1955 N.Y. Times 22 May vi. 19/2 Maybe it's all these new buildings breeding more of these cool Brooks Brothers cats. 1955 Sci. News Let. 1 Oct. 221/2 This is not cool chatter between some young hep-cats in a smoke-filled jazz joint. 1957 Sunday Mail (Glasgow) 10 Feb. 11 Gone—the best, in the top rung, the coolest. 1958 Observer 23 Nov. 16/3 On one side was the frenetic+bumptiousness of the rock-'n'-rollers, on the other the calculated indifference of the cool cats. 1959 Ibid. 25 Oct. 29/8 They got long, sloppy haircuts and wide knot ties and no-press suits with fat lapels. Very cool.