@MoscowFleet - I agree! The Trinity is not easy to understand, and you're spot on in the critique of the water/ice/steam example. It's not a good analogy because the Trinity is (3 persons in 1 essence), while water can be (1 essence in 3 forms). "Forms" aren't the same as "persons." In my own understanding, I've found it helpful to learn some of the specific terminology involved. Of course, everyone who has had Phi 101 thinks they know enough, and everyone with a bachelor's in philosophy is going to think they know everything. You can't trust the self-proclaimed "experts" on what is right, because most of the time the nature of truth and right and existence is the hotly contested subject of discussion anyway.
As I understand it, the distinction between an essence (what a thing is in itself, it's nature, often called a "substance"), and a person (a rational, individual substance (philosophical substance, not matter-based substance)) are distinct. Generally we only see one person per substance, because we generally only see finite and discrete substances like ourselves having persons. Yet God's substance/nature is infinite and creative, so the natural (and necessary) outpouring of that substance is the eternal begetting of the Son and the procession of the Holy Spirit as distinct persons within that same nature. Because the divine essence has the property of self-giving Love, He also then acts to create and sustain the world which is external to Him - thus creating other substances (like us!) to receive that Love.
The Trinity math doesn't work out if we're speaking of finite substances (1+1+1=1), but that operation is perhaps an oversimplification. What about (1*1*1=1) or (1^1^1=1) ? Or how many (1/infinity)'s are there between 0 and 1? Using mathematics may not be the most precise language if we're dealing with philosophical or theological concepts, but I'm game for trying.
Anyway, those are my meager thoughts for a beginning, and I'm glad for the opportunity to share and discuss. Here's some things I've found helpful as I have been driven to learn:
https://www.ccel.org/creeds/athanasian.creed.html
The Athanasian Creed is a good example of defining the relationships between the three persons and the one divine essence. fwiw, this creed isn't designed to be an argument, but a statement of the conclusions. Check it out though, because it does help clarify what church bodies mean when they speak about the Trinity. (your mileage with an individual's understanding of that teaching may vary)
http://www.newadvent.org/summa/1.htm
Thomas Aquinas explains it pretty thoroughly here, even if the text might be somewhat inaccessible. Questions 27-43 are of particular interest to this thread, and I remember liking the analogy made in Q93p5 about the image of God in man.
(!) https://www.youtube.com/watch?v=KQLfgaUoQCw
If you're looking for a hilarious video instead of hours of reading, check out this video where St. Patrick talks to two farmers about bad analogies for the Trinity.