For heinous say: /ˈheɪ.nəs/ like greyness
— ɴᴏᴛ /ˈhaɪ.nəs/ like dryness
— ɴᴏʀ /ˈhiː.ni.əs/ like Phineas
The Canadian speaker in the first clip is using the phoneme we refer to as the PRICE vowel. This phoneme is by convention normatively written as the diphthong /aɪ/ in both Received Pronunciation and General American, but the exact phonetics for that phoneme vary by dialect and speaker. He appears to saying [ɐi] there. This is a non-standard pronunciation of heinous according to the OED. He should not be using the PRICE vowel there.
In contrast, both the American and the Australian in the second clip are each using the phoneme which we call the FACE vowel. This phoneme is by convention normatively written as the diphthong /eɪ/ in both Received Pronunciation and General American, but as with all phonemes, the exact phonetics for that phoneme vary by dialect and speaker. The American appears to saying [eɪ] there, and the Australian perhaps [ɛɪ]. Those are two allophones of the same phoneme, so exactly which gets used phonetically will not change which word one hears said. This is a perfectly standard pronunciation of heinous according to the OED because it uses the FACE vowel.
In other words, heinous should rhyme with greyness and anus, ɴᴏᴛ with dryness and sinus.
Quoth the OED
The OED gives a primary pronunciation for heinous in both the UK and the US alike with the first syllable pronounced like hay as in hayfields, the so-called FACE vowel. However, it also gives an alternate, secondary UK pronunciation whose first syllable is like hee as in tee hee, which is the FLEECE vowel. The OED presents no evidence that heinous is ever pronounced with a first syllable like high as in high school, which is the PRICE vowel.
Historical spellings dating from Middle English include both heynous and hainous. Its etymon, however, is French haineux, in Old French haïnos, haïneus, from haine hatred, from ha-ïr meaning to hate. So the “high” pronunciation was historically there in Old French and perhaps in Middle English, but never that I can see in modern versions of either language.
I have heard native speakers of English from the Appalachia region of the United States use the "highness" pronunciation, and like the OP found this peculiar. I have yet to uncover any documentation in support of this variation.
However, others have noticed this and called it wrong. For example, there’s a blog post by V. J. Singal from 2011 that calls this the mispronounced word of the week:
This past week, while speaking about the tragedy in Tucson, numerous Americans have used the word “heinous” [pronounced HAY-nus] to describe Jared Loughner’s crime. No surprise there. What has amazed me is the number of prominent and well-educated people who have mispronounced that word, including University of Arizona President Robert Shelton. In his opening remarks at the nationally broadcast memorial service on Wednesday night (the one attended by President Obama), Shelton pronounced heinous as “highness.” Unbelievable!
And in the PBS column “Beastly Mispronunciations”, they cite The Big Book Of Beastly Mispronunciations: The Complete Opinionated Guide For The Careful Speaker by Charles Harrington Elster, who offers this opinion:
This word is so frequently mispronounced by well-educated speakers (who ought to know better) that you’d think the dictionaries would have caved in by now and countenanced the blunder. To their credit, they have not. HAY-nis is the only acceptable pronunciation of this word. Do not say HEE-nis or HEE-nee-is (HEE- as in heat), both of which are HAY-nis and long-standing members of the Most Unwanted Beastly Mispronunciations Hit List. Heinous has two syllables, not three, and the first syllable is pronounced like the word hay, never like he. See mischievous, grievous, intravenous.