As user6726's answer mentions, IPA letters (like [i], [e], [ɪ], [ɨ], [a]) refer to ranges of sounds (phones); for vowel letters, these can be thought of as regions of the "vowel space" (which in turn can be thought of as either an acoustic space (defined by certain frequencies), or a physiological space (defined by tongue position)).
And phonemic transcriptions (conventionally indicated by the use of slashes rather than brackets) don't even technically use IPA letters to refer directly to the associated ranges of phones. Rather, traditions for phonemic transcription establish conventional associations between IPA symbols and particular phonemes in a language, regardless of the actual phonetic realization of the latter. If there gets to be too much of a difference between the conventional transcription and the phonetics of some particular accent, a new convention may be proposed/adopted, but often the old conventions still stick around for a long time. You can see some of the diversity in transcriptions of English vowels on the web page "IPA transcription systems for English" by John Wells.
The phonetic realization of the phoneme /aɪ/ is variable between accents, and between different utterances made by a single speaker (some of the variation can be predicted fairly easily from the context, and some can't).
Mark Liberman, a linguist, has a post on Language Log mentioning some of the features that may make a phonetic transcription like [aɪ] valid for the phoneme /aɪ/ in at least some contexts in some present-day accents of English: The rɑɪt sɑʊnz?
But it isn't necessarily incorrect to use something like /aj/ as a phonological transcription. Geoff Lindsey, a linguist and dialect coach, transcribes the standard Southern British English "price" vowel as /ɑj/. In his blog post "The British English vowel system", Lindsey says:
in SSB, the upper corners of the vowel space function as the
endpoints of its two subclasses of diphthongs. I prefer to transcribe
them as the semivowels j and w, which are exactly equivalent in terms of the vowel space, but have additional benefits. First, j and w make explicit that the English diphthongs are falling, ie moving from
greater to lesser prominence. Second, they reinforce English vowel
linking for the foreign learner, eg way out wɛjawt, show off ʃəwɔf, as
well as initialisms like those in the previous paragraph.
Additionally, they provide by far the best way to teach cardinals 1
and 8 to native English speakers: simply pronounce intervocalic j or
w, then “hit the pause button” in the middle of the semivowel.
In final position, especially when heavily accented, SSB diphthongs
are sometimes followed by a short, schwa-like sound which highlights
the final j or w. Here are me and loo said by Kate Winslet: [sound sample on the original blog post]
Such pronunciations clearly differ from the closing diphthongs of old
RP with their distinctively not-quite-high endpoints ɪ and ʊ. These
sound very old-fashioned, especially in final accented position, e.g.
May in this 1932 newsreel clip: [sound sample on the original blog post]
or the Queen saying Christmas Day in 1957: [sound sample on the original blog post]
or the narrator of the 1948 film Quartet saying 24 plays: [sound sample on the original blog post]
Of course, in the less accented contexts of continuous speech the
contemporary j and w endpoints will often not be reached. This kind of
target undershoot is a universal phenomenon in running speech.
My summary:
There are phonetic reasons for the traditional transcription of the vowel of PRICE as /aɪ/; some of them are historical but others continue to be relevant. The use of this transcription however does not necessarily imply that /aɪ/ ends in the same sound as the KIT vowel phoneme on either a phonological or a phonetic level.
There is nothing wrong with using another transcription like ai, aj or even ay if you have some valid reason to do so. The choice of symbols for phonemes in a phonemic transcription is basically arbitrary, and most phonemic analyses treat the "PRICE" vowel as a single phoneme.
Transcriptions that identify the last element of the PRICE with the palatal semivowel (as in the start of the word "yam") have some theoretical support and are also supported by a tradition of use that continues to the present day, although it is not as prominent as the tradition that uses /aɪ/.
The FLEECE vowel of course is also phonetically and phonologically similar to the palatal semivowel, but I think transcriptions like ai have not been as popular as ones like aj because using a symbol like j that is explicitly dedicated to transcribing glides makes it clear that we are not talking about a disyllabic sequence.
I'd recommend not worring too much about the details of phonemic transcription unless you're planning on learning or teaching some kind of standardized transcription system (e.g. for use in EFL contexts). Whether you're primarily interested in phonology or in phonetics, IPA transcriptions are fairly superficial*, and the most commonly used transcriptions of phonemes tend to be common because of tradition rather than because of deep theoretical reasons.
*I don't mean this in a negative way. If IPA transcription were more strongly based on phonological theories, it would make it harder to give transcriptions in a relatively theory-neutral fashion; if IPA transcription were more strongly based on phonetic details, it would make it harder to give "broad" transcriptions that are appropriate for describing how a speaker pronounces a phoneme across a range of contexts, or how a group of speakers tend to pronounce a phoneme. But the compromises of IPA mean that there is often more than one theoretically reasonable way to transcribe something, and so the specific way that people actually do transcribe it is based on tradition and somewhat arbitrary from a theoretical perspective.
