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The cardinal vowel No.4 [a] pronounced by Daniel Jones and some other linguistics sounds more like /æ/ as in cat.
but this cardinal vowel pronounced in the IPA website(by 4 speakers) sounds more like /ɑ/ as in father,
what is the correct way to pronounce this vowel?
The four corner vowels have a special place, because they were defined by Jones in terms of "extremes". Other vowels are (poorly) defined in terms of auditory interpolation between physically-fixed positions. IMO, Esling and House are closer to each other in those productions of [a, æ], and Ladefoged is most divergent. We have single tokens of each vowel: we would want multiple tokens, in order to factor out random speaker variation.
One factor that should be considered in understanding expert performances is "actual language experience". It is well-known that the vowel conventionally transcribed as [a] is actually produced quite differently across languages. Actual (extensive) experience with "a" in Italian can influence one's pronunciation of IPA [a]. The vowel [a] really represent a range of possible formants, not a single precise point, likewise the less-clear vowel [æ]. Ladefoged's performance of [æ] is distinctly more-American (West Coast) sounding, which should not be surprising given where he lived. If we had recordings of Catford's pronunciations, we could get a fifth opinion.
Some variability may be because of vocal tract properties of a particular performer, but the main source of variability is the small set of defining reference samples. For example, as far as we know, there is exactly one set of Ladefoged IPA recordings. By convention, the IPA (organization) gets to sanction particular performances and ignore others. It might be of interest to you to read Ladefoged's (1959) dissertation, where he collects measurements from expert performers who were "closer" to Jones, which may enable you to get a better understanding of the range covered by IPA vowels (with less passage of time since initial training).
