How I would look at it is that there is a process available to English speakers of vowel reduction in certain phonological environments, such as in the syllable immediately preceding the stressed syllable. Certainly, there are differences between speakers in how much and how often they reduce pre-tonic vowels. In fact, I hear a diphthong in the American’s pronunciation in the link provided by aloofsocialite, so we can see that this is not a simple matter of “British” vs. “American” varieties (few things truly are).
I am arguing that /aɪ/ is the only unreduced pronunciation, and there are also reduced forms, with the level of reduction depending on the speaker. This would mean that apparent instances of the sound [ɪ] would in fact be the realization of this reduced vowel. It may seem academic to distinguish between the /ɪ/ that shows up in stressed syllables and the reduced "schwi" vowel that we see here. However, one prediction of the model I’m presenting is that the [ɪ] sound would only pop up in this word in contexts where vowel reduction is possible. That is, I would predict that speakers like Angela Rippon, if asked to say each syllable one-by-one, would produce ɔ.gan.aɪ.zeɪ.ʃʌn or perhaps ɔ.gan.aɪz.eɪ.ʃʌn rather than ɔ.gan.ɪz.eɪ.ʃʌn. But I don’t actually know-- perhaps some speakers do in fact have the [ɪ] sound in all contexts for this word.
[Note: another reason it can be useful to distinguish the reduced vowel "schwi" from a "short i" is that schwi can be represented by other letters than i, such as the e in "equator".]
