I am going to have to disagree with the other answers.
Of course in isolation or in carefully enunciated speech, I is pronounced "eye". But in rapid, everyday, casual speech, you will find that the "I" is, as you describe, a lighter sound that blurs into the previous vowel. The phrase "May I take a seat?" will come out "May-ah tak-ah seat?" (where "ah" represents a reduced vowel or schwa). In extreme examples, the "I" might not even be pronounced at all.
There are two issues here: first, most native speakers don't notice this phenomenon. We know exactly what's being said, so our brains fill in the sound we expect to hear, which is a nice, long, independent "I". Variations on the same phoneme (called allophones) tend to get filtered out in our brains when the sound difference does not make a distinction between words in our native language.
Second, this pronunciation difference is situational. Different regional dialects, and even different speakers, in different contexts & at different times, will show this phenomenon to a greater or lesser extent. I tend to talk fast, so I do it often. Someone who speaks more slowly probably won't show it as much in their speech. Also, the vowel reduction will go away when someone is speaking carefully and clearly--which makes it even harder for native speakers to realize it's happening, because when they stop to listen, they pronounce things carefully. It also means you are less likely to see it in actors, who tend to enunciate more when reading their lines. Your video "May I speak to Cathy?" does show this phenomenon (your ears are not wrong!), but it's actually not nearly so strong here as it usually is, precisely because he's an actor reading a line and being careful to pronounce it correctly. (Well, and also, the character is trying to be respectful to the father of a girl he's calling--so he would likely take some pains to sound as clear and "proper" as possible.)
Finally, you mentioned that you feel people don't understand you. I can't say whether that's true, but I do not believe most English speakers would have an issue understanding "I" pronounced "eye" in this context. (It is what they expect.) There may be some other issue in your pronunciation of these words, but we can't really diagnose that through the Internet!