A likely reason why the "Y" names don't sit well with you, Annalisa, is because they are commonly used by Hebrew Roots cultists who want to be more "authentic" by calling God by His Hebrew name, and Jesus by His. Even some people who are not Hebrew Roots cultists sometimes like to use Hebrew words or Hebrew names for God, because they think that doing so makes them more "Christian" somehow. Some Pentecostals and Charismatics do this, for example. There's an element of Judaising with this as well (like what happened in New Testament times). And as "Aussie Chris" has just mentioned, people also tend to use the Hebrew names of God as kind of "magic" words. They think that God will grant their wishes if they address Him by His "proper" Hebrew names.
If they really want to be more authentic, they should call Jesus "Iesous", the Greek name which was used in the New Testament (originally written in Greek). "Yeshua" is actually a Hebrew translation of this since the name Jesus doesn't appear in the Old Testament. (It can also be pronounced "Yashua" and the name "Joshua" is closely associated with it.) The word Christ is from the Greek "Christos", which means "anointed one". (Fun fact: the Hebrew equivalent is "Mashiach", which is where we get our word "Messiah". In Psalm 2:2, this word is used for the English word "anointed", and when the verse is quoted in Acts 4:26, the word "Christos" is used in the Greek text, which is why the English translation says "Christ" rather than "anointed" again.)
The letter "J" is a rather curious one. In many languages, it's pronounced like "Y". (These include German, Dutch, Scandinavian languages and some central European languages.) In others, it's pronounced more like our J, but often softer (like in French). In some languages, like Greek, their equivalent of "I" doubles as a "J", and is pronounced like "Y" at the beginning of words. So "Iesous" is pronounced "YE-soos". The Hebrew "alef-bet" includes a little letter called yod (also spelt yodh). It looks like a small apostrophe: '. (Another fun fact is that in the "alef-bet", there are three letters that look like vertical lines. I call them "The Three Bears". The "papa bear" is Nun - the Hebrew N - which takes one shape at the beginning or in the middle of words, but looks like a large vertical line at the end of a word. The "mama bear" is Vav, which is a medium vertical line and is the equivalent of either B or V, depending on circumstances. It can also be used to substitute for vowels, so ends up being used a lot in modern Hebrew. And the "baby bear" is Yod. So if you ever decide to learn Hebrew, that's an easy way to remember those particular three letters.) When used at the start of words, it gives them a "y" sound. Hebrew words starting with Yod tend to become either J or Y words in English. For example, in Hebrew, the word "Jehovah" is pronounced like "Yehova" and starts with this Yod letter. Why we say "Jehovah" and not "Yehova" probably has to do with the way the English language has developed through the centuries.
A note of caution on the name "Jah": while this name is Biblical, it is used by the Rastafarians as the name for their god. They have stolen it from Psalm 68:4. Obviously, using that name in prayer doesn't make you a Rastafarian, but you wouldn't want people to think that if you happened to be praying with others around. Also, the name "Jehovah" has been hijacked by the Jehovah's Witnesses, so be a bit cautious with that as well.
In conclusion, there is nothing wrong per se with using Hebrew names for God, but there's also absolutely no need for it. The names of God and Jesus Christ are perfectly fine for all speakers of English. As I say, the Hebrew names are commonly used by people who are making a pretence of being more "authentic Christians" by going back to the Bible's "original language". But if they truly wanted to be more authentic, they should incorporate Greek as well as Hebrew names into their vocabulary. (Of course, that would be just as pretentious.) To me, the most Biblical address we can use in prayer is simply "Father" or "Heavenly Father". Jesus Himself prayed to the Father, and the Lord's Prayer (which was meant as a general model for prayer and not for repetitious prayer heathen-style) opens with the words "Our Father". However, if you want to use another Biblical name for God when praying, I don't think that's wrong. The Bible never says that God won't hear us if we don't use "the right name", but it does say He won't hear us if we regard iniquity in our hearts (Psalm 66:18). So He doesn't hear the people who use His Hebrew names for a pretence, because He can see the pride in their hearts behind that.
Interesting about the Invictus Games using "I Am" in their logo. They were founded by Prince Harry, who has all sorts of issues, but this is not the thread to go into those. I'll just say that it doesn't surprise me that something he's behind would include this sort of thing. Incidentally, I have a smattering of modern Hebrew and Greek through studying them on Duolingo. They differ somewhat from the ancient languages used to write the Bible, just as modern English differs from old or medieval English, but anyway, that's how I have come to know a bit about them.
