From CantoDict Dictionary:
(jyutping) cit3, (pinyin) qie1
 cut; slice; chop; mince; carve
 (indicates that the previous two characters are 反切 notation)
 [math] tangent
(jyutping) cit3, (pinyin) qie4
 correspond to; conform to; accord with
 close to; near to
 eager; anxious; keen
 be sure to; must
(jyutping) cai3, (pinyin) qie4
[1a ]all; whole
From the question:
to be close to
(Match pinyin /qie4/ ) close to
to bite tightly
only work in idiom 咬牙切齒, which literally means 'bite and cut with teeth, therefore, 切 itself doesn't contain the meaning of 'bite' only contains the meaning of 'cut' here
切 doesn't have the meaning of to 'clench' . The editor might have confused by the expression '切齒', which literally means 'cut with teeth' and this action results in you shut your jaws tight, thus extend to the meaning of 'clenching jaws'
to correspond to
(Match pinyin /qie4/  correspond to)
to match (traditional Chinese medicine) to feel the pulse
to feel the pulse is 搭脈; 切脈 means 'hack the blood vein' which is a martial art term, the editor made a mistake again
(Match pinyin /qie4/  eager)
(Match pinyin /qie4/  keen)
only work in compound word '急切' which means ' urgently close'. '切' in '急切' doesn't mean urgent, it means 'close' or 'near'; only '急' in '急切' means urgent
only work in compound word '逼切' which means 'pressingly close'. '切' in '逼切' doesn't mean 'pressing', it means 'close' or 'near'; only '逼' in '逼切' means 'pressing'
(Match pinyin /qie4/ [1a])but it only work in compound word '一切', '切' itself doesn't mean 'all' - Even the CantoDict dictionary made the same mistake
(Match pinyin /qie4/ [1a])- 'all' extended to mean 'entire'
(Match pinyin /qie4/  must)
by all means
the editor was confused by the expression '不顧一切' which literally means '不顧'(not care), 一切 (all); Only the idiom '不顧一切' means 'by all means', '切' doesn't
'main point' in Chinese is '要點', the editor might have confused by a common phrase '切中要點' which literally means '切中'(cut correctly into) 要點(the main point).
(phonetics) Short for 反切 (fǎnqiè, “fanqie”).
(Match (pinyin) qie1 )
(chiefly women's speech, colloquial) An interjection conveying a slight sense of disdain or dissatisfaction.
'啐' sounded like '切'. It is a phonetic loan word for an exclamation that express disdain or dissatisfaction