In some books, the long "a" found in words like say, play, etc. are transcribed as /sei/, /plei/ respectively but in some others the same words are transcribed /se/ and /ple/. Which one is correct? Can we consider both? What is the standard IPA transcription of the long "a"?
In purely phonetic terms, it is most definitely not an [e]. Few dialects of English have a true [e] (some have [eː], but that’s a different sound).
The ‘long a’ diphthong can be variously transcribed as [ɛɪ], [ɛi], [eɪ], or occasionally [ei], depending on how broad the phonetic transcription is, what dialect is being transcribed, and a host of other factors. (Various diacritics to show more precisely a specific pronunciation may also be added.)
In phonemic transcription, which is what I think you’re really asking about (despite the phonetics tag), it is usually transcribed /eɪ/ or /ɛɪ/, since phonemic transcriptions of English most commonly assume that the underlying difference between [ɪ] and [iː] is one of tense-/laxness, rather than length: they are thus phonemicised as /ɪ/ and /i/, which would make /ei/ a combination of lax (short) [e/ɛ] following by tense/long [iː]—a combination I don’t think exists in any natural English word.
As most phonemicisations of English do not recognise a similar phonemic opposition between lax (short) /ɛ/ and tense (long) /e/, it is purely a matter of convention and habit whether you write the (lax/short) phoneme as /e/ or /ɛ/: both are in common use.
I don’t believe I have ever come across any literature that does recognise a tense (long) /e/ as a phoneme and identifies it with the diphthong [ɛɪ], though doing so would not be particularly heretic—long /o/ is a corresponding diphthong [oʊ] or [əʊ], after all. This would likely break down once you start comparing different dialects of English, though—various vowel mergers in dialects means that ‘English’ phonemes are really more like supraphonemes.